A Catalog of the Real Numbers - Carnegie Mellon...
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A Catalog of the Real Numbers
by Michael Ian Shamos, Ph.D., J.D.
School of Computer Science Carnegie Mellon University
2007
Copyright © 2007 Michael Ian Shamos. All rights reserved.
Preface
Have you ever looked at the first few digits the decimal expansion of a number and tried to recognize which number it was? You probably do it all the time without realizing it. When you see 3.14159... or 2.7182818... you have no trouble spotting and e, respectively. But are
you on familiar terms with 0.572467033... or 0.405465108...? These are 2 3
12
and
, both of which have interesting stories to tell. You will find them, along with those of more than 10,000 other numbers, listed in this book. But why would someone bother to compile a listing of such facts, which may seem on the surface to be a compendium of numerical trivia?
log log3 2
Unless you had spent some time with 2 3
12
, you might not realize that it is the sum of two
very different series, ( )cos
12
1
k
k
k
k and . The proof that each sum
equals
1
)12()2(k
kkk
2 3
12
is elementary; what connection there may be between the two series remains
elusive. The number log turns up as the sum of several series and the value of certain
definite integrals:
log3 2
1
31k
k k
, ( )
1
2
1
1
k
kk k
,
)2)(1 xx
dx
1 ( and
0 2 xe
dx
1 . If your research led to
either number, this information might well lead you to additional findings or help you explain what you had already discovered.
Mathematics is an adventure, and a very personal one. Its practitioners are all exploring the same territory, but without a roadmap they all follow different paths. Unfortunately, the mathematical literature dwells primarily on final results, the destinations, if you will, rather than the paths the explorers took in arriving at them. We are presented with an orderly sequence of definitions, lemmas, theorems and corollaries, but are often left mystified as to how they were discovered in the first place.
The real empirical process of mathematics, that is, the laborious trek through false conjectures and blind alleys and taking wrong turns, is a messy business, ill-suited to the crisp format of referred journals. Possibly the unattractiveness of the topic accounts for the scarcity of writing on the subject. How many unsuccessful models of computation did Turing invent before he hit on the Turing machine? How did he even generate the models? How did he realize that the Halting Problem was interesting to consider, let alone imagine that it was unsolvable? What did his scrap paper look like?
This book is a field guide to the real numbers, similar in many ways to a naturalist’s handbook. When a birdwatcher spots what he thinks may be a rare species, he notes some of its characteristics, such as color or shape of beak, and looks it up in a field guide to verify his identification. Likewise, when you see part of a number (its initial decimal digits), you should be able to look it up here and find out more about it. The book is just a lexicographic list of 10,000 numbers arranged by initial digits of their decimal expansions, along with expressions whose values share the same initial digits.
PREFACE iv
This book was inspired by Neil Sloane’s A Handbook of Integer Sequences. That work, first published in 1971, is an elaborately-compiled list of the initial terms of over 2000 integer sequences arranged in lexicographic order. When one is confronted with the first few term of an unfamiliar sequence, such as {1, 2, 5, 16, 65, 326, 1957,...}, a glance at Sloane reveals that this sequence, number 589, is the total number of permutations of all subsets of n objects, that
is,
n
k
n
k
n
k jn
kn
n
k
nk
111 !
1!
)!(
!! , which is the closest integer to n . Sloane’s book is
extremely useful for finding relationships among combinatorial problems. It leads to discoveries, which help in turn generate conjectures that hopefully lead to theorems.
e!
What impelled me to compile this catalog was an investigation into Riemann’s -function that I began in 1990. It struck me as incredible that we have landed men on the moon but still
have not found a closed-form expression for ( )31
31
kk
. I say incredible because in the 18th
century Euler proved that ( )sk s
k
1
1
is a rational multiple of s for all even s, and gave a
formula for the coefficient in terms of Bernoulli numbers. No such formula is known for any odd s. It is not even known whether a closed form exists.
I began exploring such sums as 1
2 k ks sk t
, 1
1 k ks sk t
and ( )
1
2
k
s sk k k t
, whose values
can often be written as simple expressions involving various values of the -function. For
example, 1 5
43
5 32 k kk
( ), 1
6
1
24 21
2
k kk
coth and the remarkable
1 1
124 22 k kk
. To keep track of these many results, I kept them in a list by numerical
value, conveniently obtained using Mathematica®. Very quickly, after experimenting with
sums of the form , I began to notice connections that led to the following
theorem.
( )ak bk
11
Theorem 1. For an integer, b an integer such that a ba 1 2 and c a positive real
number, then 1
1 1ck kbj
aj b
ca bk
j
( )
For example, when I observed that the sums
.
An proof is given at the end of the chapter. The proof is elementary, but I never would have been motivated to write down the theorem in the first place without studying the emerging list of expressions sorted by numerical value.
After a time, I became somewhat obsessed with expanding the list of values and expressions and found that as it grew it became more and more useful. I began to add material not directly related to my research and found that it raised more questions than it answered. It became for me both a handbook and a research manifesto.
1
221
k
k
and ( )2
4 11
kk
k
of the form
shared the same 20-
digit prefix, this evoked considerable interest, since sums1
0 abk
k
are very difficult
PREFACE v
to treat analytically. After verifying that the sums w ual to ov
look for generalizations, exploring such sums as
ere eq er 500 digits, I began to 1
230
k
k ,
1
440
k
k ,
( )2
1
k
21k
k and
( )3
4 11
kk
k
give a few examples. The results suggested a conjecture, which stheorem:
, to
oon yielded the following
Theorem 2. For c > 1 real and p prime, 1
11 1c cpi
kp
kp
k
i
(, where
) ( )n is the
unction.
For a proof, see the end of the chapter. Alternatively, the theorem may be expressed in the
form
Moebius f
1
1
1
)(
kka
kp
1 /1kpp
k
a . This allows us to relate such strange expressions as
12
1k and
k 3
1k to Moebius function sums. Unfortunately, the theorem seems to
1 3k e
shed no light on trickier sums like 12
1 kk
k
and 1
1 k kk
k .
This b what is kna small space a potentially huge number of objects. It is similar to the drawers in the card catalog of a library. Imagine that a library only has 26 card drawers and that it places catalog
them. nd a b k by Taylor, you must go to drawer “T” and
ook is own in computer science as a hash table, a structure for indexing in
cards in each draw without sorting To fi oo look through all of the cards. If the library has only a hundred books and the authors’
names are reasonably distributed over the alphabet, you can expect to have to look at only two or three cards to find the book you want (if the library has it). If the library has a million books, this method would be impractical because you would have to leaf through more than 20,000 cards on the average. That’s because about 40,000 cards will “hash” to the same drawer. However, if the library had more card drawers, let’s say 17,576 of them labeled “AAA” through “ZZZ,” you would have a much easier time since each drawer would only have about 60 cards and you would expect to leaf through 30 of them to find a book the library holds (all 60 to verify that the book is not there).
Hashing was originally used to create small lookup tables for lengthy data elements. The term “hashing” in this context means cutting up into small pieces. Suppose that a company has 300 employees and used social security numbers (which have nine digits) to identify them. A table large enough to accommodate all possible social security numbers would need 109 entries, too large even to store on most computer hard disks. Such a table seems wasteful anyway, since only a few hundred locations are going to be occupied. One might try using just the first three digits of the number, which reduces the space required to just a thousand locations. A drawback is that the records for several employees would have to be stored at that location (if their social security numbers began with the same three digits). An attempt to store a record in a location where one already exists is called a “collision” and must be resolved using other methods. There will not be many collisions in a sparse table unless the numbers tend to cluster strongly (as the initial digits of social security numbers do). Using this scheme in a small community where workers have lived all their lives might result in everyone hashing
PREFACE vi
to the same location because of the geographical scheme used to assign social security numbers. A better method would be to use the three least-significant digits or to construct a “hash function,” one that is designed to spread the records evenly over the storage locations. A suitable hash function for social security numbers might be to square them and use the middle three digits. (I haven’t tried this.)
Getting back to the library, notice that any two cards in a drawer may be identical (for multiple copies of the same book) or different. The only thing they are guaranteed to have in common is the first three letters of the author’s last name. Now imagine that the library sets up 2620 drawers. Aside from the fact that you might have to travel for some time to get to a dra
authors have the same name, even if their nam
wer, once you arrive there you will probably find very few cards in it, and the chances are excellent that if there is more than one card, you are dealing with the same author or two authors who have identical names. Of course, it’s still possible for two authors to have different names that share the same 20 first letters.
This book is a hash table for the real numbers. It has 1020 drawers, but to save paper only the nonempty drawers are shown. But it differs very significantly from the card catalog of a library, in which the indexing terms (the “keys”) have finite length. When you examine two catalog cards, you can tell immediately whether the
es are very long. When you look up an entry in this book, you can’t tell immediately whether the expressions given are identically equal or not. That’s because their decimal expansions are of infinite length and all you know is that they share the same first 20 digits. With that warning, I can tell you there is no case in this book known to the author in which two listed quantities have the same initial 20 digits but are known not to be equal. There are many cases, however, in which the author has no idea whether two expressions for the same entry are equal, but therein lies the opportunity for much research by author and reader alike.
As a final example, seeing with some surprise that sin sink
k
k
kk k
1
2
21
1
2
led me to an
investigation of the conditions under which the sum of an infinite series equals the sum of the squares of its individual terms. (We may call such series “element-squaring.”) Element-squaring series rarely occur naturally but are in fact highly abundant:
Theorem 3. Every convergent series of reals B bkk
1
such that B bkk
( )2 2
1
can
be transformed into an element-squaring real series
B bk ing to B a single real by prependk 0
term b0 iff B B( )2 1 . Proof: see the end of the cha
I hope that these examples show that the list offers both information and challenges to the reader. It is a reference work, but also a storehouse of questions. Anyone who is fascinated by mathematics will be able to scan through the pages here entrie for a ber are related.
4
and wonder if, how and why the s given num
ts. The reader is encouraged to develop his own sup
pter.
The selection of entries in this book necessarily reflect the interests and prejudices of the author. This will explain the relatively large number of expressions involving the -function. For this I make no apology; there are more interesting real numbers than would fit in any book, even listing only their first 20 digi
plements to this catalog by using Mathematica to develop a list of numbers useful in his or her own field of study.
PREFACE vii
Professor George Hardy once related the now-famous story of his visit to an ailing Ramanujan in which Ramanujan asked him the number of the taxicab that brought him. Hardy thought the number, 1729, was a dull one, but Ramanujan instantly answered that it was the smallest integer that can be written as the sum of two cubes in two different ways, as 1 123 3 and
integer such that
9 103 3 . Hardy marveled that Ramanujan seemed to be on intimate terms with the integers and regarded them as his personal friends1. It is hoped that this book will help make the real numbers the reader’s personal friends.
Proofs of Theorems 1-3
Theorem 1. For a 1 an integer, b an a b 2 and c a positive real
num( )b
c k
aj
ckjj
a
1 1
1at t the chapter.
ber, then kb b . A proof is given he end of
Proof: The left-hand sum may be written as the double sum 1
11aj b j
kj
k c
order of summation,
. Reversing the
1111
11 11 1 k jk jj k
1
1jabjajbjbaj
ckkckkck , which equals
1 1
1
1
1 1k ck ck kbk
a a b bk
p
. QED.
Theorem 2. For c > 1 real and prime, 1
11 1c
kp
cpi
kpk
i
( ), where ( )n is the Moebius
function.
Proof: ( ) ( )kp kp
kp jkp
. Note thc ck jk 11 11
at every term of 1
i
1 c pi
sum, certainly in the case when
appears at least once in
the ouble 1 and j pik d . We will show that the sum of all
coefficients of 1
c pi in the double sum is zero except when k 1 is a power of p, in which and j
case ( )kp 1, and the result follows. Suppose that jk is a power of p. Then k ( )kpif 1, 0 since kp has a repeated factor.
Therefore, only the terms with k 1 contribute to the sum and ( )kp 1 each j a power of p. Now consider all pairs j, k with jk not a power of p. Suppose that i has the factorization i
p a1 th each of the a 0. pq , wi i If aq1 p k or if k has any repeated prime factors, then
( )kp 0. Since k i, it suffices to consider only those k whose factors are products of subsets P P o set f distinct primes taken from thek ppp q 1 , a set having cardinality q 1 or q,
on whether depending P contai If ns p. Pk as an odd number of elements, then h ( )kp 1; , if otherwise Pk has an even number of elements, then ( )kp 1. ( )kp 0 cannot be zero
since p does not divide k. But the number of subsets of Pk having an odd num
ber of elements
1This incident is partially related in Kanigel, R. The Man Who Knew Infinity. New York: Washington Square Press (1991). ISBN 0-671-75061-5. The book is a fascinating biography of Srinivasa Ramanujan.
PREFACE viii
is ing an even numbeble sum
the same as the number of subsets hav r of elements. Therefore, the terms cancel each other and terms with jk not a power of p do not contribute to the dou . QED.
Theorem 3. Every convergent series of reals B bkk
1
such that B bkk
( )2 2
1
can
be transformed into an element-squaring series
B bkk 0
by appending to B a single real term
b0 iff B B( )2 1
4 .
Proof: For B to be element-squaring, it is necessary that
b b b b B b Bkk
kk
2 2 2 2 2
( ) . Solving this quadratic equation for b yields 00
01
0
0
bB B
0
21 )1 4
( )(
2, which is real iff B B( )2 1
4 0 . Since is finite, b B b B0
that if
converges. It is also eleme y to showntar B B( )2 1
4 , then there is no real ele ent that m
can be prepended to B to m ke it element-squaring. QED.
a
Note that B does not necessarily imply that B( )2 b. For example, let kk
k
( )1
.
Then
2
1,
1B , which is finite, but 2
Bkk
( )2 1
1
, which ( )2Bdiverges. Likewise, does
not imply that . If bkk 1
, then B( )2
6
2 but B diverges. B
Using This Book
This book is a catalog of over 10,000 real numbers. A typical entry looks like:
.20273255405408219099... log log3 2
2
1
5
arctanh AS 4.5.64, J941, K148
= 1
5 2 1
1
22 10
1
11
kk
k
kkk k
( )
( )
=
12)12(
log
x
dxx
= dxx
xxx1
0
5sinsin)( GR 1.513.7
Each entry begins with a real number at the left, with its integer and fractional parts separated. Entries are aligned on the decimal point. Three dots are used to indicate either (1) a non-terminating expansion, as above, or (2) a rational number whose period is greater than 18. Three dots are always followed by the symbol , which denotes “approximately equal to.”
The center portion of an entry contains a list of expressions whose numerical values match the portion of the decimal expansion shown. The entries in the list are separated by the equal sign =; however, it is not to be inferred that the entries are identically equal, merely that their decimal expansions share the same prefix to the precision shown.
There is no precise ordering to the expressions in an entry. In general, closed forms are given first, followed by sums, products and then integrals. At the right margin next to expression may appear one or more citations to references relating the expression to the numerical value given or another expression in the entry. When more than one expression appears on a line containing a citation, it means that the citation refers to at least one expression on that line. See also, “Citations,” below.
Format of Numbers
Numbers are given to 19 decimal places in most cases. Where fewer than 15 digits are given, no more are known to the author. Rational numbers are specified by repeating the periodic portion two or more times, with the digits of the last repetition underlined. For example, 1/22 would be written as .04504545, which is an abbreviation for the non-terminating decimal .0454545454545045... . Since rational numbers can be specified exactly, the = symbol rather than is used to the right of the decimal. Where space does not permit a repetition of the periodic part, it is given once only. If the period is longer than 18 decimals, the sign is used to denote that the decimal has not been written exactly. No negative entries appear.
Ordering of Entries
Entries are in lexicographic order by decimal part. Entries having the same fractional part but different integer parts are listed in increasing magnitude by integer part. Rational numbers are treated as if their decimal expansions were fully elaborated. For example, these entries would appear in the following order:
USING THIS BOOK x
.505002034457717739... .5050150501 = .505015060150701508... .5050 = .505069928817260012...
Arrangement of Expressions
In general, terms in a multiplicative expression are listed in decreasing order of magnitude except that constants, including such factors as ( )1 k appear at the left. For example, 4 2 3k kk! is used in preference to 4 2 . Additive expressions are ordered to avoid initial minus signs, where possible. For example, 3
3k kk !2 k is used instead of 2 3k . These principles are
applied separately in the numerators and denominators of fractions.
Citations Abbreviations flushed against the right margin next to an expression denote a reference to
an equation, page or section of a reference work. Page numbers are prefixed by the letter p, as in “p. 434.” A list of cited works appears at the end of the book.
Acknowledgments
No book like this one has previously appeared, probably because of the monumental difficulty of performing the required calculations even with computer assistance. However, with the development of the Mathematica® program, a product of Wolfram Research, Inc., it has been possible to obtain 20-digit expansions of most of the entries without undue difficulty. The symbolic summation capability of that program has been invaluable in developing closed-form expressions for many of the sums and products listed. Even though my involvement with computers extends back almost 35 years, I have no hesitation in declaring that Mathematica is the most important piece of mathematical software that has ever been written and my debt to its authors is, paradoxically, incalculable.
The tendency to be fascinated with numbers, as opposed to other, abstract structures of mathematics, is not uncommon among mathematicians, though to be afflicted to the same degree as this author may be unusual. If so, it because of the role played by a collection of people and books in my early life. My grandfather, Max Shamos, was a surveyor who, to the end of his 93-year life, performed elaborate calculations manually using tables of seven-place logarithms. The work never bored him — he took an infectious pride in obtaining precise results. When calculators were developed (and, later, computers), he never used them, not out of stubbornness or a refusal to embrace new technology, but because, like a cabinetmaker, he needed to use his hands.
I read later in Courant and Robbins’ What is Mathematics, of the accomplishments of the calculating prodigies like Johann Dase and of William Shanks, who devoted decades to a manual determination of to 707 decimal places (526 of them correct). At the time, I couldn’t imagine why someone would do that, but I later understood that he did it because he just had to know what lay out there. I expect that people will ask the same question about my compiling this book, and the answer is the same. The challenge set by Shanks is still alive and well — every year brings a story of a new calculation — is now known to several billion digits.
My father, Morris Shamos, is an experimental physicist with a healthy knowledge of mathematics. When I was in high school, I was puzzled by the formula for the moment of
inertia of a sphere, M m r2 . That is was proportional to the square of the radius made
sense enough, but I couldn’t understand the factor of 2/5. Dad set me straight on the road to advanced calculus by showing me how to calculate the moment by evaluating a triple integral.
2
5
As a teenager during the 1960s, I worked for Jacob T. (“Jake,” later “Jack”) Schwartz at the New York University Computer Center, programming the IBM 7094. (That machine, which cost millions, had a main memory equivalent to about 125,000 bytes, about one-tenth of the capacity of a single floppy disk that today can be purchased for a dollar.) NYU had a large filing cabinet with punched-cards containing FORTRAN subroutines for calculating the values of numerous functions, such as the double-precision (in the parlance of the day) arctangent. By poring over these cards, I learned something of the trouble even a computer must go through in its calculations of mathematical functions. It was this exposure that drew me to field that later came to be called computer science.
ACKNOWLEDGMENTS xii
For the past 20 years I have been an enthusiastic user of Neil Sloane’s A Handbook of Integer Sequences in combinatorial research, so it is particularly pleasing to note that this book itself consists entirely of such sequences.
My greatest debt, however, is to Stephen Wolfram for his creation of Mathematica, the single greatest piece of software ever written. In developing this catalog, I used Mathematica on and off over a 15-year period to explore symbolic sums and integrals and to compute most of the constants that appear in this book to 20 decimal places.
Notation
a n( ) greatest odd divisor of integer n.
Ai x( ) Airy function, solution to y xy 0. AS 10.4.1
b x( ) ( )
1
0
k
k x k.
Bk kth Bernoulli number, B Bk k ( )0 . AS 23.1.2
B xk ( ) kth Bernoulli polynomial, defined by te
eB x
t
k
xt
t k
k
k
1 0
( )!. AS 23.1.1
bp n( ) number of divisors of n that are primes or powers of primes.
ck coefficient of xk in the expansion of ( )x .
C x( )
0
2
142
0
2
)14(2)!2(
)1(cos
2
1)(
kk
kkkx
kk
xdttxC
. AS 7.3.1, 7.3.11
Ci x( ) cosine integral,
x
k
kk
kk
xxdt
t
txxCi
0 1
2
)2()!2(
)1(log
1coslog)( .
AS 5.2.2, AS 5.2.16 d n( ) number of divisors of n, d n n( ) ( ) 0
dn kth derangement number, d d d d1 2 3 40 1 2 9 , , , , nearest integer to k e!/ . Riordan 3.5
E x( ) complete elliptic integral
dxxE 2/
0
2sin1)( .
En nth Euler number,
2
12 k
kk EE . AS 23.1.2
E xn( ) nth Euler number, defined by 2
1 0
e
eE x
t
k
xt
t k
k
k
( )!
. AS 23.1.1
Ei x( ) exponential integral
1 !
log)(k
k
x
t
kk
xx
t
dtexEi . AS 5.5.1
Erf x( ) error function
0
12
0
12
0 !)!12(
2)1(2
)12(!
)1(22 22
k
kkkx
k
kkxt
k
xe
kk
xdte
. AS 7.1.5
Erfc x( ) complimentary error function = 1 Erf x( ). AS 7.1.2 f n( ) number of representations of n as an ordered product of factors
Fn nth Fibonacci number, F F F F Fn n n1 2 11 2 ;
0 1F (; ; )a x hypergeometric function x
k a
k
kk ! ( )
0
, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
1 1F ( ; ; )a b x Kummer confluent hypergeometric function x a
k b
kk
kk
( )
( )!
0 ,
NOTATION 599
where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
2 1F ( ; ; ; )a b c x hypergeometric function x a b
k c
kk k
kk
( ) ( )
( )!
0
, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
Catalan’s constant ( )
( ). ...
1
2 10 915965594177219015052
0
k
k k G
Gn
1 )13(
1
)13(
11
knn kk
gn
1 )13(
1
)13(
11
knn kk
gd z( ) Gudermannian, gd z ez( ) arctan 22
Hn harmonic number: Hkn
k
n
1
1
H ne even harmonic sum: H
ke
n
k
n
1
21
H no odd harmonic sum: H
ko
n
k
n
1
2 11
j( )n generalized harmonic number: H
kj
n
jjj
n jk
n j j( )
( )
( )( ) ( )
( )!,
1 1 1
11
1
1 1
H
hn
0 )56(
)1(
)16(
)1(
kn
k
n
k
kk
hg x( ) Ramanujan’s generalization of the harmonic numbers, hg xx
k k xk
( )( )
1
I xn( ) modified Bessel function of the first kind, which satisfies the differential equation
z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9
jn
1 )13(
1
)13(
11
knn kk
J311
J xn( ) Bessel function of the first kind ( )
!( )!
1
2
2
20
k n k
n kk
x
k n k LY 6.579
k 1 2 k
K x( ) complete elliptic integral of the the first kind
dx
2/
02sin1
1
K xn( ) modified Bessel function of the second kind, which satisfies the differential equation
z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9
l x( )
n
kk k
k
xk
xk
k
k
11
log)log(log , when x is an integer n.
NOTATION 600
li x( ) logarithmic integral x
t
dtxli
0 log)(
Li xn( ) polylog function Li xx
kn
k
nk
( )
1
log x natural logarithm loge x
a prime number. denotes a sum over all primes. pp
pd n( ) product of divisors pd n dd n
( )|
pf n( ) partial factorial pf nkk
n
( )!
1
0
pfac n( ) product of primes not exceeding n, pfac n pp n
( )
pq n( ) product of quadratfrei divisors of n, pq n pr n( ) ( ) 2
pr n( ) number of prime factors of n
r n( ) 4 ( )|
dd n HW 17.9
s n( ) smallest prime factor of n
sd n( ) sum of the divisors of n, sd n n( ) ( ) 1
S x( ) 34
1
2
0
2
)34()!12(
)2/()1(
2sin)(
k
k
kkx
xkk
dtt
xS
. AS7.3.2, AS 7.3.13
S n m1( , ) Stirling number of the first kind, defined by the recurrence relation:
x x x n S n m xm
m
n
( )...( ) ( , ) 1 1 1
0
. AS 24.1.3
S n m( , )2 Stirling number of the first kind, nm
k
km kk
m
mmnS
02 )1(
!
1),( . AS 24.1.4
Si x( ) sine integral,
x
k
kk
kk
xdt
t
txSi
0 0
12
)12()!12(
)1(sin)( . AS 5.2.1, AS 5.2.14
T s( ) 1
ksk , where k has an odd number of prime factors
v1
2
23)31(
2
23)31(1
ii
iiv
w x( ) w x e erfc ixix
kx
k
k
( ) ( )( )
( / )
2
2 10
Wn
0 )56(
)1(
)16(
)1(
kn
k
n
k
kk
z x2 ( ) z x k xx
x xk
k2
2
12
11( ) ( ( ) ) ( ( ))
NOTATION 601
( )s ( )( )
( )s
k
k
sk
1
2 1
1
0
Euler's constant
n k
n
knlim log
1
1
0.57721566490153286 AS 6.1.3
f
n
kn
dxxfkf1 1
)()(lim
( )s Riemann zeta function 1
1 ksk
( )m ( )s derivative of the Riemann zeta function ( log )
k
k
m
sk 1
( )s ( )( )
( )sk
sk
s( )
k
s
1
1 21
1
1
( )s ( )( )
( ) (sk
ss
)k
s
1
2 11 2
0
( )n Moebius function, ( ) , 1 1 ( ) ( )n k 1 if n has exactly k distinct prime factors, 0 otherwise. ( )n number of different prime factors of n.
k n( ) sum of kth powers of divisors of n, dk
d n|
golden ratio, 1 5
21 61803397498948482
. ...
( )n Euler totient function, the number of positive interegs less than and relatively prime to n.
( )n polygamma function ( )nkk
n
1
1
1
( )x kth derivative of the polygamma function ( ) ( )( )k
k
kx
d x
dx
( )n ( ) ( )n kk
n
1
( )s )()1(
21)( 2/ ss
ss s
( )n overcounting function ( )n -1 + the number of divisors of the g.c.d of the exponents of the prime factorization of n.
.00000000000000000000 = ( ) ( )
!
( )
!
1 1 1
0
1 2
1
k
k
k
k
k
k
k
k GR 1.212
= ( ) ( )
!
( )
!!( )
1 2 2 11
2
20 1
2
1
k k
k
k
k
kk
k
k
k
k
k
k
k
k
= ( )k
kk
1
Berndt Ch. 24 (14.3)
= ( )k
kk 2 11
= ( )sin
1 13
1
k
k
k
k
= log2
20 1
x
x
1 .000000000000000000000 = 1
11 k kk ( )
J396
= 1
2 21 kk
½
= k
kk ( ) !
11
GR 0.245.4
= 2
20
k
k
k
k( ) !
= 1
10 ( )!k k!k
= 2 1 3 3 1
3 3
4 3
72
2
6 5 41
k k
k k
k k
k k k k3k k
= 2 1
4 1
2 1
2 11 1
k
k k
k
k kkk k
( )
( )!
( )! ( )
!
=
( ) logk k
kk 1
= F k
kk
4 2
1 8
=
22 22k
k
kk
H
= ( )kk
12
=
2
)(
k k
k
=
( ) logk k
kk 1
Berndt Ch. 24, Eq. 14.3
= 1
1
S
, where S is the set of all non-trivial integer powers
(Goldbach, 1729) JAMA 134, 129-140(1988)
=1
21 n
k
k
n
kn !
log
!
= ( )sin
121
1
k
k
k
k
= 11 1
1
( )k
k k J1065
= dx
x
x dx
x21
21
log
= dx
x( ) /2 30 1
2
= dx
x x( )1 1 20
1
= x x
edxx
log( )1
0
=
cossin
xx
x
dx
x0
Berndt I, p. 318
= arctan x
xdx2
1
2 .000000000000000000000 = (½, , )!
( )! ( )!
!
1 02
2
2 11 1 1
k k
k
k
kkk
k
k k
= ( )
( )
k H F
k
k
kk
kk
kk
kk
k
1
2 2
1
4 1
2
1 1 0
=
1
1)2(exp
k k
k
=
12
)(
k k
kz
= 11
2 11
2
22
k k
k
kk k( )
= 11
220
k
k
= e ek kk
k
k
k
( ) / /
1
1
1 2
1
1
= log
(cos )
2
21
2
20
2
1
x
xdx
x
edx
dx
o
= x x
xdx
sin
cos
/
10
2
GR 3.791.1
= logsin cos/
x x d2
0
2
x
= e dxx
0
1
=
0
11
3xe
dxx
3 .000000000000000000000 = k k
kkk 6
2 1
0
4 .000000000000000000000 = k k k
k
k
kkk
kk
kk2
1
2
1
4 1
2 21
1 2 0
( )
( )
= ( )!!
( )!! ( )
( )!!
( )!! (
2
2 1
2
2 122
121
k
k k k
k
k k kk k
)
6 .000000000000000000000 = (½, , )
2 02
2
1
kk
k
= k
k kk
2
3 2 2( )( )
= x dx
e
xdx
xx
3
0
3
21
log
10 .000000000000000000000 = ( )k F k
kk
kk
k
4
2 20 1
12 . 000000000000000000000 = ( )
log
k dx
x xxe dxk
k
x
1
2
2
0 2 0
15 . 000000000000000000000 = ( / , , )1 3 4 03
4
1
kk
k
20 . 000000000000000000000 = k
k kk
2
4 3 3( )( )
24 . 000000000000000000000 = log4
20
xdx
x
26 . 000000000000000000000 = (½, , )
3 02
3
1
kk
k
52 . 000000000000000000000 = ( )k
kk
1
2
3
0
70 . 000000000000000000000 = k
k kk
2
5 4 4( )( )
94 . 000000000000000000000 = F kk
kk
2
1 2
150 . 000000000000000000000 = (½, , )
4 02
4
1
kk
k
1082 . 000000000000000000000 = (½, , )
5 02
5
1
kk
k
1330 . 000000000000000000000 = F kk
kk
3
1 2
25102 .000000000000000000000 = F kk
kk
4
1 2
1 .00000027557319224027... 1
101 ( )k !k
... 697
6613488
8
8
W J313
...
648000, the number of radians in one second
1 .000006974709035616233... 2coth
... 10
1 .0000115515612717522... 547 61
2 3 5
1
6 5
1
6 1
7
10 9
1
7
1
71
( )
( )
( )
( )
k k
k k k J328
1 .000014085447167... W7 J313
.0000248015873015873... 13
1 ( )k
!k
... 9
... sec1
.00004439438838973293... 1
22528
11
121
23
12
1
4096 12 1 122
Fdx
x, , ,
... e10
1 .0000513451838437726... ( )
( )
( )9
511 9
512
1
2 1 90
kk
AS 23.2.20
1 .00005516079486946347... 1
358318080
1
12
5
12
7
12
11
125 5 5 5 ( ) ( ) ( ) ( )
= ( )
( )
( )
( )
1
6 5
1
6 1
1
6
1
61
k k
k k k
1 .0000733495124908981... W6
691
87480
J313
... 8
... e9
...
( ) ( )log3
3
21
2k
kk
1 .000155179025296119303... ( )
( )
( )8
17
161280
255 8
256
1
2 1
8
80
kk
AS 23.2.20
1 .0002024735580587153... 3
31
.0002480158734938207... 1
81 ( )k !k
...
1
8192
1
16
1
16 12
30
( )
( )kk
.0002499644008724365... 16 6 112 6
2 1k kk
k k
( )
1 .0002572430074464511... 5 61
2 3
1
6 5
1
6 1
7
7 6
1
5
1
51
( )
( )
( )
( )
k k
k k k J328
.00026041666666666666 = 1
3840
1
5 25
!
.0002908882086657216...
10800, the number of radians in one minute of arc.
.000291375291375291375 =1
34321
492
8
kk
...
( )
( )( )3
4096
1
81921
1
162
31
kk
... 7
... e8 1 .00038992014989... W5 J313
1 .000443474605655004... ( )
( )
( )
( )
1
4 3
1
4 1
1
7
1
71
k k
k k k J326
1 .0004642141285359797... 1
7776
1
6
1
6 13
41
( )kk
1 .00047154865237655476... ( )
( )
( )7
127 7
128
1
2 1 70
kk
AS 23.2.20
1 .00049418860411946456... ( )11111
1
kk
1 .00056102267483432351...
1
3456
1
12
1
12 112
31
( )
( )kk
1 .000588927172046... r( )16 Berndt 8.14.1
5
82
1 2
4 2
2 2
16
2 2 2
2 2 2
2 2
16
2 2 2
2 2 2log
log( )log log
... 11
22
21
24
9
23
15
25
1
2
1 3
51
( ) ( ) ( ) ( )( )
( )
k
k
k
k
.00069563478192106151... ( )
( )
3
1728
1
12 31
kk
... ( )( )
12
21
2
1
kk
k
k k
.00088110828277842903...
1
3456
11
12
1
12 12
31
( )
( )kk
.00091188196555451621... e7
1 .00097663219262887431... 15
1 k kk
1 .00099457512781808534... ( )10110
1
kk
.00099553002208977438... ( )10 11
kk
.00101009962219514182... 13
124 6 4 1
1
110 4
2
( ) ( )kk kk k
.00102614809103260947... ( )5 5 11
110 5
2
kk kk k
.001040161473295852296... 6
.0010598949016150109... 15
8 42 6 4 6 1
1
110 6
2
coth ( ) ( ) ( )k
k kk k
.00108225108225108225 =1
9241
362
7
kk
.00114484089113910728...
1
3456
5
6
1
12 22
31
( )
( )kk
1 .00115374373790910569... 1
124416
1
12
5
12
7
12
11
123 3 3 3 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
6 5
1
6 1
1
4
1
41
k k
k k k
1 .001242978120195538...
1
1458
1
9
1
9 82
31
( )
( )kk
1 .00130144245443407196... 1
31457280
1
8
3
8
5
8
7
85 5 5 5 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
4 3
1
4 1
1
6
1
61
k k
k k k
.001311794958417665... 197
27 64
3 3
32
1
2 8 30
( ) ( )
( )
k
k k
.00138916447355203054... e
e kk4
1
41
1
2
1
4 21
cos
( )!
3 .00139933814196238175... 2
1 .0013996605974732513... 3 3
62
3
4
3 1
312
log
log log( )( )r Bendt 8.14.1
.00142506011172369955... 1
102 k kk log
.00144498563598179575... ( )2
62
k
kk
.00144687406599163371... ( )2
162
k
kk
1 .0014470766409421219... 6
61960
663 6
64
1
2 1
( )( )
( )kk
AS 23.2.20
...
1
8192
9
16
1
16 92
30
( )
( )kk
... ( ) ( )log4
4
21
2
k
kk
.00153432674083843739...
1
3456
3
4
1
12 32
31
( )
( )kk
.0015383430770987471... 930 5
1920 2 1
6
61
( )
( )
k
kk
.0015383956660985045... 4 3
256
1
4 8 30
( ) ( )
( )
k
k k
1 .001712244255991878...
1
1024
1
8
1
8 72
31
( )
( )kk
1 .00171408596632857117... 5 3
6
( )
... 1
1536
1
4
1
4 13
40
( )
( )
kk
.001815222323088761... 5
54
197
216
1
1 2 3
2
2 2 21
2
k k k kk ( ) ( ) ( ) LY 6.21
.001871372759366027379... 7
360
3
2
1
1
3
3 21
( )
k e kk
Ramanujan
=
32
1 1
( )k
e kk
.001871809161652456210... 1
12 21
22
1k e
k
ekk
kk
( )
.001872682449768546116... 1
121
12
1k e
k
ekk
kk
( )
.00187443047777494092... 1
1 121
02
1e
k
ekk
kk
( )
.001912334879124743... 15 109
960
2
6
2
0
e
k
k
k
( )!
1 .001949804343... j9 J311 1 .00195748553... J309 G9
2 .001989185473323053... 1110
1
kk
1 .00200839282608221442... ( )9
.00201221758463940331... ( )9 11
119
2
kkk k
.00201403420628152655... 3
46
3
25
1
23
5
44
9
83
17
162
387
642 ( ) ( ) ( ) ( ) ( ) ( )
= H
kk
k
15
1 2( )
.00201605994409566126... ( )8 1 11
19
2
kk kk k
.00201681670359358... 3277
3375 32
1
2 7
3
30
( )
( )
k
k k
.00202379524407751486... ( )7 2 11
19 2
2
kk kk k
.002037712107418497211... )7()3()5()10(4
7)1,9( 2 MHS
.00203946556435117678... ( )6 3 11
19 3
2
kk kk k
... 7 3
4096
1
8192
1
2
1
16 82
30
( )
( )( )
kk
.00210716107047916356... ( )5 4 11
19 4
2
kk kk k
.0021368161356763007...
1
3456
2
3
1
12 42
31
( )
( )kk
.00213925209462515473... ( )4 5 11
19 5
2
kk kk k
1 .0021511423251279551... 5
486
4
4
W J313
... 19
11520
2
96
2
48
3 2
8
5 3 2 4
log log ( ) log
=1
4 2 1
25
0k
k k
k
k( )
.00227204396400239669...
1
1458
8
9
1
9 12
31
( )
( )kk
1 .00277832894710406108... cosh cos1 1
.00228942996522361328... 1
3 6 19 6
1 1k kk
k k
( ( ) )
.00234776738898358259... ( )
( )
3
512
1
8 31
kk
... log
logsin/2
4 2 0
4
G
x dx GR 4.224.2
=
log/ x
xdx
1 20
1 2
GR 2.241.6
1 .0024250560555062466... 2 2
5
5
4
3
2 5
1 5
210
log loglog ( )
r Berndt 8.14.1
.002450822732290039104...
dx
e
exx
x
2
4
5 14
3
.00245775047043566779... 171109
24 5
2
1
ek
kk
( )!
.002478752176666358... e6
2 .0025071947052832538... 6 2 8 2 62 2
22
0
log log( )
k H
kk
kk
.00260416161616161616 =1
284
1
4 24
!
... 1
2
1
2
1
2
1
4 4
1
2
1
2
1
4 160
cos cosh cos( )!
e
e
k kk
... 1
8
5
3
1
4
1
4 7 12
log arctan
dx
x x
... dx
x72 1
... loglog
263
6 72
dx
x x
... dx
x x7 22
... 1
4
5
3
1
8 7 32
log
dx
x x
... cosh cos1 1
... loglog
27
3
1
24 7 42
dx
x x
.00286715218149708931... 1
92 k kk log
...
1
3
1
)5(
)1(
4
)3(3
216000
195353
k
k
k
=
1
0
1
0
1
0
555
1dzdydx
xyz
zyx
5 .0029142333248880669... 2
10
k
k k !
.002928727553540092... ( ) ( )
( )
3
36 324
1
3
2 1
31
k
k
k
k
...
1
8192
7
16
1
16 72
30
( )
( )kk
.0031244926731183736...
1
3450
7
12
1
12 52
31
( )
( )kk
.00318531972162645303... 2
33
2 33 2
2ex
x x
sinsin
... loglog
23
2
9
24 7 52
dx
x x
.00326776364305338547... 5
...
1
1458
7
9
1
9 22
31
( )
( )kk
... Re ( ) i
.0033104481564221302... 1
16
7
45 256
4
.0033178346712119643... 3 3
32
7
64
1
2 6 30
( ) ( )
( )
k
k k
=
1
0
1
0
1
0222
555
1dzdydx
zyx
zyx
24 .00333297476905227... e e e e
e e
k
ekk
( )
( ), ,
3 2
5
4
1
11 11 1
1
14 0
... 15 5
16
31
32
1
3 50
( ) ( )
( )
k
k k
.00337787815787790335...
1
1024
7
8
1
8 12
31
( )
( )kk
.003472222222222222222 =1
288 1 2 3 4 51
k
k k k k kk ( )( )( )( )( )
.00364656750482608467... 1
27
3
36
1
3 3
1
31
( ) ( )
( )
k
K k
=
1
0
1
0
1
0333
555
1dzdydx
zyx
zyx
...
3
2
36 3
91 3
216
1
432
1
6
( ) ( ) Berndt 7.12.3
= 1
6 5 31 ( )kk
.0037145966378051377... log( log( log(log ))) 2 1 .00372150430231012927... log( log(log ))2
.00374187319732128820... coth
12
12e
1 .00374187319732128820... coth
e e
e e
=3
800
1
2 3 41
5
( )( )( )(k k k k )k
... 19
2 2
1
4 3
1
4 1
5
12
1
51
1
5
( )
( )
( )
( )
k
k
k
k k J 326
1 .003795666917135695...
4 3 2 5
8 2 1 40
2arctan log dx
x
.00388173171751883... 32
256
1
4 2
3 1
31
( )
( )
k
k k
... 22
72
2
3
1
2 1
3
0
log( )
( )
k
kk
k
k
1 .0038848218538872141... arctan2
... j8 J 311
... 14
1 k kk
... 656
6200145
8
8
G J 309
.00396825396825396825 =1
2525 1
252
6
( )kk
.00402556575171262686... 7 4
8
7
8 4
18 4
2
( )
sinh
( )
k
k k k
=( ) ( ) ( )
int
1 1
142
4
1
k
k a non trivegerpower
k
k
ial
.004031441804149936148...
02
2
3 18
1dx
e
exx
x
.00406079887448485317... 19450 8 1
88
( )
( )
1 .0040773561979443394...
8
94508 ( )
.00409269829928628731... 15
16 8
2
8
2 2
2 2
cothsin sinh
cos cosh
= ( )8 11
118
2
kkk k
... 10
... 17 1 18
2 1k kk
k k
( )
... 3
42
2 3
3
2
( ) tanh
= 16 2 18 2
2 1k kk
k k
( )
.004147210828822741353... 1
12 52 ( )kk
... dxx
x
1
0
22
1
arccos
2421616
.004166666666666666666 =1
2407 ( )
.004170242045482641209... )6()3()5()4()7()2()9(4)1,8( MHS
... 15 3 18 3
2 1k kk
k k
( )
... 15
84
4 ( ) coth
= 14 4 18 4
2 14
1k kk
k
kk k k
( )( )
.00442764193431403234... ( )
( )( )
( )
2
2
7
1803 3 1
1
2
4 1
41
2
k
k
k
k
... 1
486
1
3
1
3 13
40
( )
( )
kk
.00450339052463230128... ( )2
52
k
kk
.00452050160996510208... ( )2
152
k
kk
...
2
2
22 2
2
82 2 1 2
cos sin csc
= k kk
kk
k k
2 2 12
21
2
2 22
( )( )
... 31 5
325
1
2 1 50
( )( )
( )
kk
AS 23.2.20
... 3
256
1
9
1
2 1 2 1 2 3
2
2 21
2
( ) ( ) (k k k )k
LY 6.21
... 12
3
64
1
2 3
3 1
31
G k
k
k
k
( )
( )
... log2661
960 7 62
dx
x x
... 13 5 18 5
2 1k kk
k k
( )
...
1
8192
3
8
1
16 62
30
( )
( )kk
... Gx
xdx
log log( )2
2
1
1
2
20
1
GR 4.295.6
=
log sin
/ xdx
20
2
... log log ( )22
41 2 8 r
.0047648219275670510... 67
64 96 360
3
8
5
2
1
2
2 4
52
( ) ( )
( )k kk
... 7
216
1
6 5
1
6 1
3 1
31
1
3
( )
( )
( )
( )
k
k
k
k k
... 7 3
1728
1
12 6 31
( )
( )
kk
.004915509943581366... sin
2
2 4
2
22
1
kk
.004983200... mil/sec
.00506504565493641652...
1
1458
2
3
1
9 32
31
( )
( )kk
.00507713440452621921... 4
51
93 5
192 2 1
( )
( )
k
kk
... 4
3
10 3
81
=kk
k
F Fk
2
12 1 2 12
1
22 2
3
2
1
4
1
33 3
5
2
1
4
, , , , , ,
...
1
3
1
)4(
)1(
1728
1549
4
)3(3
k
k
k
=
1
0
1
0
1
0
444
1dzdydx
xyz
zyx
.0051783527503297262... 7 3
128 512
1
1024
3
4
1
8 2
32
31
( )
( )( )
kk
.0053996374561862323... 15
42 4 6 2 6
1
( ) ( ) ( ) ( )kk
1
=1
8 62 k kk
=1
180 2 3 4 51
k
k k k kk ( )( )( )( )
6 .005561414313810887... 2
2 23
2
0log
x
dxx
... ( , )5 3
1 .00570213235367585932... 1
2
2
3
3
4
1
216 631
log log
k kk2 Ramanujan] Berndt Ch. 2
... 1
81 k kk log
.005899759143515937... 45 7
8 6
( )
Berndt 9.27.12
1 .005912144457743732... 1
250
1
5
1
5 42
31
( )
( )
kk
1 .0059138090647037913...
3
1 33
243
91 3
486
1
324
1
6
1
17496
1
6
( ) ( ) ( )
=( )
( )
2 1
6 1
2
40
k
kk
... 3
3032
26
27
1
2 5
( )
( )
k
k k
.00603351696087563779... log k
k 7
...
1
4096
7
162
2
8 81
( ) log x
x xdx
...
1
3375
7
152
2
8 71
( ) log x
x xdx
...
( ) log( )3
196
1
2744
1
22
2
8 61
x
x xdx
...
2 )(
)1(
)1(
)(
k k
k
k
k
... k
kk2
1 1
...
1
2197
7
132
2
8 51
( ) log x
x xdx
...
1
1728
7
122
2
8 41
( ) log x
x x
...
1
42
2
16
5 2 2
5 2 2arctan log
( )
arctan arctanlog1 2
8
2
81 2 1 2
3
8
2
16 6 22
dx
x x
31 .0062766802998201758... 3 2
0 1
log( )
xx
x xdx GR 4.261.10
= x e
edx
x
x
2 2
1
/
GR 3.4.11.16
.00628052611482317663... 2 1
3148
2
8
3 3
32
1
2 2
2
log ( ) ( )
( )
k
k
k
k
00629484324054357244... 1
160
5
61
11
6
1
64 12 1 62
Fdx
x, , ,
...
1
1331
7
112
2
8 31
( ) log x
x xdx
... log log21
531 6
2
dx
x x
= 4
6251
41
3
1
( )kk
k
k
...
1
1000
7
102
2
8 21
( ) log x
x x
.006476876667430481... log arctan3
4
2
2
1
2 4 6 22
dx
x x
... e e e
e e
k
ekk
2
4
3
0
4 1
1
13 0
( )
, ,
... 5
3
...
1
729
7
92
2
8 11
( ) log x
x xdx
... ( ) ( )i i
... log log
( )
2
4
1
6 1 50
1
x
xdx
...
( )2 1
2 11
kk
1
k
... e5
...
1
512
7
8 12
2
81
( ) log x
xdx
... ( )( )
( )4 1
12
422
24
1
jk
f k
kji k
.00677885613384857755... 3
46 5
1
23 4 3 22 ( ) ( ) ( ) ( ) ( ) ( ) 5
= H
kk
k
15
1 1( )
... log
arctan7
6
1
3
5
3
3
6
1
8 6 32
dx
x x
... 2
1
28
11
108
arctan
...
962 2 8 2 12 3
1
4 2 1
22 3
40
log log ( )( )
kk k
k
k
... 2 3
343
2
81
( ) log
x
x xdx
.00702160271746929417... 7
48 180
3 3
4
23
8
1
1 2
2 4
41
( )
( )( )k k kk
... 16
3
128 1
3 2
2 31
Gx
xdx
log
( )
... 9
... 3
8
1 1
4
1
1
e k
k
k
( )
( )!
1
... ( ) ( )
1 21 4
1
k
k
k
1 .007160669571457955175... 1
230 kk
... 1
2454 3 2 452 ( )
.0071918833558263656... i i ii i cos( log ) sin( log )
...
1
216
7
62
8 21
( ) dx
x x
2 ...
1
216
1
62
1
216
7
6 12 2
2
60
1
( ) ( ) log x dx
x
... sech2
1
( )kk
... 17
360
3 3
4
1
2
4 1
41
( ) ( )
( )
k
k
k
k
... Gk
k
k
10016
11025
1
2 7
1
21
( )
( )
... 1
6
1
22
1
csch ( )k
k
[Ramanujan] Berndt Ch. 26
= 1
1329 ( )
.00763947766738817903... log3
2
13
24 6 42
dx
x x
... j7 J 311
... 1
71 ( !)kk
... 1716 15428
15
2
135
1
12
4 6
7 71
k kk ( )
... 119
72 6
1
1 2
2
2 21
3
k k k kk ( ) ( ) ( )
1 .00788821... J 309 G7
...
1
125
7
52
2
8 31
( ) log x
x xdx
... 3 5
44
4 4
( )( ) c Berndt 9.27.12
.008062883608299872296...
dx
e
exx
x
2
2
3 14
1
.00808459936222125346... 4
1 22
2
0
2
log logx e x dxx
...
1
8192
5
16
1
16 52
30
( )
( )kk
2 .00815605449274531515... 11 1 1 14
1 8 3 8 5 8 7 8
sin( ( ) sin( ( ) sin( ( ) sin( ( )/ / / /
= 11
81
kk
... 1
614496 1536
1
4
3
43 3 3
G ( ) ( )
... ( )( )
3 13 3
32
f k
kk
Titchmarsh 1.2.14
...
1
3456
5
12
1
12 52
30
( )
( )kk
.00828014416155568957... ( )
( )
7 1
7
... 1
2
1
3
2
3
3
2 3
1
3 2
1
1
e
e
k
k
k
cos( )
( )
!
1 .00833608922584898177... sin sinh sin
( )
1 1
2
1
2 4
1
4
1
4 10 !
e
e kk
... 1
81 1
1
8 41
cos sin( )
e
k
kk 1 !
1 .00834927738192282684... ( )7
=
0
624
)42)(32)(22)(12(4
)2(
381
64
381
)5(64
1143
)3(8
kk kkkk
k
=
0
236
)72)(62)(52)(32)(12(4
)2()1175793942604248(
7560 kk kkkkk
kkkk
Cvijović 1997
... ( )8 1 11
17 1
2
kk kk k
.008650529099561105501... )11(2
331)9()2(84)7()4(21)6()5(4)4,7( MHS
... 4
... ( ) log( )6 1 1 1
1
6
2
k
k
k
kk k
... 1
4
2
3
1
3
1 3
2
1 3
2
i i
= 16 1 17
2 1k kk
k k
( )
.0085182264132904860...
1
1458
5
9
1
9 42
31
( )
( )kk
... 15 2 17 2
2 1k kk
k k
( )
.008650529099561105501... )5()3(7560
)1,7(8
MHS
.00866072400601531257...
1
1024
5
8
1
8 32
31
( )
( )kk
...
1
64
7
4
1
64
3
4
2
272 2
2
8 41
( ) ( ) log x
x xdx
... 7
83
2
1
4
( ) ( )i i
= 13 4 17 4
2 1k kk
k k
( )
1 .00891931476945941517...
( )
( )
( )6
7 71
k
kk
Titchmarsh 1.2.13
5 .00898008076228346631... 1
20 ( )!kk
.00898329102112942789... i i3
... 91 3
216 36 3
1
432
5
6
1
6 1
32
31
( )
( )( )
kk
.009197703611557574338... 3
8 4 3 3 3
4 1
91
cot coth
( )kk
k
...
02 )7)(4(!
2)1(
4
7
4
52
k
kk
kkke
... 3 3 23
1 12
21
2
31
( ) log ( )
( )
k
k
k
k
... 5
3
2
3 90
1
3
3 3
2
3 3
2
4
i i
= 13 4 17 4
2 1k kk
k k
( )
... 7
45 16
15
16
7 4
8
15
16
1
3
4
40
( ) ( )
( )
k
k k
2 .00956571464674566328...
161
4
643
41
22
40
1Gx dx
x
log
...
96 2 2
1
8 7 8 5 8 3 8 11
( )( )( )(k k k kk )
J243
.00960000000000000000 = 6
625
3
61
log x
xdx
... ( )
( )
3
125
1
5 31
kk
... 64
9
8
)3(
=
1
0
1
0
1
0222
555
1dzdydx
zyx
zyx
... 7
108 1
3 2
60
log x
xdx
... 5
2
... 9 3
961
2 2
21
31
( )( )
( )
k
k
k
k
.0099992027408679778... 1
3072
1
4
3
4 63 3
3
( ) ( )
4
... 2
33
1
6
1
320
csch
( )k
k k
... i i i i2
4 2 2 21 1 2 2 ( ) ( )
i
2
2
2 2coth
=H
kk
k2 1
21
... 3 3
2
1549
864
2
6 51
( ) log
x
x xdx
... 4
...
1
27
7
32
2
8 51
( ) log x
x xdx
... 69
16
3
8
3
2
1
1 2
2
2 31
( )
( ) ( )k k kk
... 7 3
16 64
1
128
1
4
32
( ) ( )
Berndt 7.12.3
=
1
1024
1
8
5
8
1
4 32 2
31
( ) ( )
( )kk
.01037718999605679651... ( )k
kLi
k kk k
1 16
26
2
1
... 11
2
1
2
1
4 4
1
1
cos cosh( )
( )!
k
kk k
GR 1.413.2
.01041666666666666666 = 1
96 1 2 3 4 5
2
1
k
k k k k kk ( )( )( )( )( )
1 .01044926723267323166... 41
41
1
16 2 21
sinh
kk
.01049303297362745807... 16
7
8
1
56
2
2
2
8 8
3
8cot
loglogsin logsin
=1
64 8
1
822 2k k
k
kk
k
( )
...
0
3)4(
)1(
4
)3(3
216
197
k
k
k
=
1
0
1
0
1
0
333
1dzdydx
xyz
zyx
... 1
24576
1
8
3
8
5
8
7
83 3 3 3 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
4 3
1
4 1
1
4
1
41
k k
k k k
.01065796470989861952... 2 5 2 33
24
5
43
9
82
81
16 ( ) ( ) ( ) ( ) ( ) ( )
= H
kk
k
14
1 2( )
.01077586137283670648... 13
8
2
4
3
12
3 1
3 1 8
3 1
3 1
log loglog
=1
12
1
12
1
12
1
121
1221 1k k
hgk
k
kk
k
( )( )
.01085551389327864275... log2131
192 6 52
dx
x x
6 .01086775003589217196... 1
256
1
4 13
3
40
1
( ) log
x dx
x
... 60
49
3600
1
2 1 2 2 2 6 2 70
( )
( )( )( )( )
k
k k k k k
... 9
43 5
17 5
2
( ) ( )k kk
= 1
1 22 5 15
1 1k k kk
k k( ) ( )( )
.01105544825889466389... 11
1536
1
4
3
4
1
2 33 3
40
( ) ( ) ( )
( )
k
k k
1 .0110826597832979188... 1
2 1 255
11 ( )
( )k k
kk k
k
1 .011119930921598195267... 1
22 2cosh cos
=
sin sin sin
( )sin
( )/ /1 11 2
2
1 2
21 4 3 4 i i
= 114 4
1
kk
... 3 12 2 9 31
12
130
log log( )( )k kk
.01121802464158449834... csc
( )cos cos sin sin
1
4 11 1 3 12
2 2
ee e
1
=1
14 41 kk
... ( )jkkj
3 115
... 1
36 6
1
8
23
0
( )k
kk
k k
k
... log log
( )3
2
2 2
36 r
... 32 2 8 2 4 3 2 4 51
2 6 51
log ( ) ( ) ( ) ( )
k kk
=( )k
kk
5
21
... 720 7 11
6
0
( )( )
x
e edxx x
... 720 7 17 ( ) ( )( )
...
3 2 3 1 1 1120
10
120
dx
x
x dx
x
... 21
71
/k
k k
...
sinh2
1
2 4
2
21
k
kk
=
1
02)1(
)log2cos(dx
x
x GR 3.883.1
... 117
2 7k ks ds
k slog( )
.01175544134736910981... coth
2
12e
...
1
125
1
5 12
2
50
1
( ) log x dx
x
.0118259564890464719... 2
1144
49
864 1 2 3 4
H
k k k k kk
k ( )( )( )( )
... 3 3
2 22 2 4
1
1 2
2 1
31
( )log
( )
( )( )
k
k
k
k k
... 3 2
6 4116
52
27
log x
x xdx
.0121188016321911204... 7
5760
5
192
93
256
1
1
4 2
2 42
( )
( )
k
k k
... e
k
k
k
2
0
7
32
2
5
( )!
.01227184630308512984...
256 162 20
dx
x( )
.012307165328788035744...
1
0
1
0
1
0222
333
132
)3(3
8
1dzdydx
zyx
zyx
.01232267147533101011... 8 1 .01232891528356646996... 5 2 6 3 ( ) ( )
.01236617586807707787... 4 2
2 41
30
768
1
2
1
4 1
( )kk
J373
.01243125524701590587... 18 13
32
9 2
32
13
128
G
log
J385
.0124515659374656125... 80
3
8
45320 2 48 3 560
1
2 1
2 4
4 41
log ( )( )k kk
.01250606673761113456... 1 4 4 8 6 3 12 52
3
61
( ) ( ) ( ) ( )( )
k
kk
.01256429785698989747... 160 44 215
24 80 2 20 3
1
2 13 41
( ) ( ) log ( )( )k kk
.01261355010396303533... 1
12 42 ( )kk
.012665147955292221430...
022 18
1dx
e
exx
x
1 .01284424247707... W3
.01285216513179572508...
( )log
66
1
k
kk
... 1
1536
3
4
1
4 33
40
( )
( )
kk
... 31 3
108
1
1 2 3
2
21
k k k kk ( )( )( )
... 3 3
44 2
24
13
4
1
1 2
2 1
3 21
( )log
( )
( ) ( )
k
k k k k
6 .01290685395076773064... 5
81 1
4 3
60
log x dx
x
... 209
225
1
2 5
1
21
Gk
k
k
( )
( )
1 .01295375333486096191... 1
23 1
21
2
1
16 4 10
log log log( )
ii
ii
kkk
... 110
2
... 1
216
4
3
5
62 2
2
6 31
( ) ( ) log
x dx
x x
1 .01321183515... ( )( )
1 12
1
k
k
k
k
... 8
1 .01330230301225999528...
0 2
12sin
2
1
2
1sin
1cos45
6
kk
k
... 7 3
4
7054
3375
1
8
7
22
2
8 61
( ) log( )
x
x xdx
... 1
43 3 12 2 2
1
12
1 3
30
( ) log( )
( )
k
k
k
k
=49
3600
1
2 1 2 2 2 6 2 70
( )( )( )( )k k k kk
...
2 4 2
4132
3
128
21
163 1
2 3
( )( )
k
kk
... ( ) ( )( )
2 6 465
8
1 4
1
2
42
k k
k kk
= ( ) ( )
1 23
1
k
k
k k 1
... cos sin( )!( )
1 11 1
4 2 10
e k kk
... 1
512
9
8
5
82 2
2
6 21
( ) ( ) log
x
x xdx
... iLi e Li e
k
ki i
k2 2 21
sin2
e
... e ie i /2
...
2
4 2coth ( ) ( ) 1
= ( ) ( )
1 2 4 111
16 4
2
k
k k
kk k
... dxxx
x
0
222
)3)(1(
log3log
12
3log
... 3 3
256
1
4 30
( ) ( )
( )
k
k k k
.01417763649649285569... 2 3
27
7
18 12 31
dx
x x( )
.0142857142857142857 =1
701
162
5
kk
2 .0143227335483157... F
kk
k !
1
5021 .01432933734541210554... 127
240 1
8 7
0
1
log x dx
x GR 4.266.1
3 .0143592174654142724... 1
1 Fkk
.01442468283791513142... 3 3
250
2
61
( ) log
x dx
x x
... e k
k
( )
1
1
1
1 .01460720903672859326... e EikH
kkk
k
12
1
2
1
2
2
21
21
log
!
.01462276787138248... ( )2
142
k
kk
1 .01467803160419205455...
4
4196
7 4
84
1
2 1
( )( )
( )kk
AS 23.2.20, J342
... 27
28
8
)3(7
=
1
0
1
0
1
0222
444
1dzdydx
zyx
zyx
... 1
1728
11
12
5
12 12 2
2
61
( ) ( ) log
x dx
x
... 2 2
12
2
1
k
k
k
k
k
ke
k
( )
!/
...
1
12
332log
4
331
8
27log
4
1
kkkH
... e
erf e x x dxx4
12
0
( ) sinh cosh
... 2 3
44
Berndt 8.17.16
... 17
16 48 180
3
4
1
2
2 4
41
( )
( )k kk
.01521987685245300483... ( ) ( ) ( ) ( )( )
4 3 6 3 5 71
3
71
k
kk
... 2 2
40648
2 1
6 3
( )
( )
k
kk
... log ( )
( )( )
2
8 16
1
8
1
4 2 4 4
1
1
k
k k k
...
2
3 21
5
43
1
1 2
( )( ) ( )k k kk
... j6 J311
...
( )36
3
1 3
6
3 3
2
1 3
6
3 3
2
i i i i
= ( ) ( )
1 3 3 111
16 3
2
k
k k
kk k
... 45 5
484 3 504 2 462
1
1
1
6 61
( )( ) log
( )
( )
k
k k k
2 .01564147795560999654... ( )8
... 1
61 ( !)kk
... 2 6 42 4 252 2 4621
16 61
( ) ( ) ( )( )
k kk
... 13
1 k kk
...
1
8192
1
4
1
16 42
30
( )
( )kk
1 .0158084056589008209... log
log log( )2
2
1
2
2 1
2
2 1
2
2
21
k
kk
... dx
x52 1
.01593429224574911842...
( )4 1
4 14 11
kk
k
1 .01594752966348281716... 104
98415
6
6
G J309
.01594968819427129848...
1
3456
1
3
243 32 3
124416
1
3456
5
32
32
( ) ( )
= 1
12 8
13 3
1728 2592 331
3
( )
( )
kk
.01600000000000000000 = 2
125
2
61
log x
xdx
1 .01609803521875068827... 1
41 k k
k
.01613463028439279292...
254
15log2log
xx
dx
.01614992628790568331... 92
32 3
1
1
2
2 32
( )( )
k
k kk
= ( ) ( )
1 31 2
1
k
k
k k 1
.0161516179237856869... 10163
6 4
2
1
ek
kk
( )!
.0162406578502357031... 31 5
8
7 3
32
2
48 2 1
2 4
41
( ) ( ) log
( )
H
kk
k
2 .01627911832363534251... 2
71
k
k k k!
1 .01635441905116282737... 1
51 k kk !
.0163897496007479593... 1 22 16
21
2 1
3
32
Gk k
k
k
log
( )
( )
... 2
6
2 2
2 2
1
2
sin sinh
= 11 4 26 2
2
1
1k kk
k
k
k
( ) ( ) 1
6
.0164149791532220206... 6 2 3 4 5 ( ) ( ) ( ) ( ) ( )
= 1 1
16 17 6
26
1 1k k k kk
k k k
( )
( )
.01643437172288507812... 7 3
512
1
8 4 31
( )
( )
kk
.0166317317921115726... ( ) ( )
13
2
k
k
k
k
.01684918394299926361... log
arctan7
6
1
2
3
6
3
3
5
3 5 22
dx
x x
.01686369315675441582... 11 5 16
2
1
1k kk
k
k
k
( ) ( ) 1
.01687709709059856862... 6155 63
1080
4 3
5 41
log x dx
x x
.01691147786855766268... 2
2112
29
36
1
1 2
3
k k k kk ( )( ) ( )
.0170474077354195802... 1945 6 1
66
( )
( )
... 13 3
216 324 3
1
432
2
3
1
6 2
32
31
( )
( )( )
kk
...
6
16
1 3
2
1 3
2coth cot
i i
12
2 2 32
2 3i icot
= 1
11 66
2
1
1kk
k
k
k
( ) ( ) 1
... 3 2 2
8
3
2 12 3
0
dx
ex /
... 6 22
4 221
2 32
log( )
k k
k
kk
k
...
6
9456 ( )
... 1
120 1
1
5
5
2
5
2
log
( ) !( )( )k
k km
k m
1 .01736033215192280239... 1
22 22 k kk
1 .01737041750471453179... sinhcosh cos
4
3 11
3 62
kk
... 18 2 16 2
2 1k kk
k k
( )
... sin1
...
180 , the number of radians in one degree
... tan1
... 17 1 16 1
2 1k kk
k k
( )
... ( )
log( )6 1
11
6
2
k
kk
k k
... ( ) ( )( )
3 3 433
16 3 41
k
kk
... 11
12 2 3
3
2
1
16 16
2 1
tanh ( )
kk
k k
... ( ) log( )5 1 1 1
1
5
2
k
k
k
kk k
... 6
2
3
2
1
2
2
6
62
sin
i
k
kk
.01765194199771948957... 2
21
8
144
1
12 3
G
kk ( )
.01772979515817195598... 1
6
11 2
18
3
4 1 21
log log
( )( )(
dx
e e e ex x x x 3)
... 1
144
1
12
5
12
7
12
11
121 1 1 1 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
6 5
1
6 1
1
2
1
21
k k
k k k
... 15 1 16
2 1k kk
k k
( )
... sinh( )
32
3
2 1
3 3 2 1
0
!
e e
k
k
k
AS 4.5.62
... ( )
( )arctan log
1
2 2 1 93
1
3
10
91
1
1
k
kk k k
... ( )5 ... e4
.018355928317494465988...
16)1(
)1,6()4()3()5()2()7(3k
k
k
HMHS
... 7
82
4
14 2 16 2
2 1
( ) coth ( )k k
kk k
.01840776945462769476... 3
512 42 30
dx
x( )
... 7 3
16 192 2 1
4
41
( )
( )
k
kk
...
2
1)()(k
kk
... 5
33
1
2
1
4 2 1 2 31
arctanhk
k k k( )( )
... ( )
( )
3
64
1
4 31
kk
... loglog
23
2
1
8 5 32
dx
x x
.01891895796178806334... log log2
3
3
4
1
30 4 3
1
36 622
k kk
=( )k
kk
1
62
1 .01898180765636222260... ( )4 1
2
1
215
1
k
k kkk k
.01923291045055350857... 2 3
125
2
61
( ) log
x
x xdx
1 .01925082490926758147...
( )
( )
( )5
6 61
k
kk
Titchmarsh 1.2.13
3 .01925198919165474986... e d
e xx
2 2 4 1
20
( )
x
.01931537584252291596... 3
4
2
3
2
22 3
3
2
1
3
1
8
2 2
2loglog
log loglog
Li
= ( )
( )
1
2 4 22
k
kk k k
.01938167868472179014... 112
3 3
2
1
2
2 1
31
( ) ( )
( )
k
k
k
k
.01939671967440508500... G k
k
k
k2 16 128
1
2 1
3 1
31
2
( )
( )
.01942719099991587856... 2
5
7
8
1 2 1
2 42
( ) ( )!
( )!
k
kk
k
k
!
.01956335398266840592... J3 1( )
.01959332075386166857... 14
96
2
32
1
2
1
2 12
1 3
21
log( )
( )Li
k
k
k
kk
.01963249194967275299... 4
33
1
61 3
3 3
21 3
3 3
2
( ) i
ii
i
= 13 3 1
1
13 16 3
2 13
2k kk
kk k k
( ) ( )
=1
13
1
a non trivialegerpowerint
306 .01968478528145326274... 5
.01972588517509853131... 1
9 108
1
3 3
2 1
21
( )
( )
k
k k
.01982323371113819152...
4
4190
17
164 3
1
2
( , )( )kk
.019860385419958982063... 3 2
4
1
2
1
64 43 21
log
k kk
[Ramanujan] Berndt Ch. 2
.0198626889385327809... 3
8
7
1920
4
5 31
dx
x x
.01990480570764553805... 3
64
3
32
3
41
31
2
1
log ( )k kk
k
k H
1 .0199234266990522067... log
log ( )5
2
1
5
1 5
25
r
1 .0199340668482264365...
2 2
2 22
2
16
5
8
1
12 2
k
kk k k
k k( )( ) ( )2
.02000000000000000000 = 1
50
... 1
22 1 30 k
kk
k
( )
.02005429455890484031... 2
9
7 2
241
1
1 141
42
log
log( )
logx
dx
x
x
x
dx
x
.02034431798198963504... 10
9
1
3
2 1
2 1
1
2 31
2
( )!!
( )!!
k
k kk
J385
.02040816326530612245... 1
49
1 .02046858442680849893... 62 5
63 61
( ) ( )
a k
kk
Titchmarsh 1.2.13
.02047498888852122466... 4 5 4 4 3 12
2
51
( ) ( ) ( )( )
k
kk
.020491954149337372830...
( )
( )
( )
( ( ) )3
3
6
1 13
14
k
k nk n not squarefree Berndt 6.30
1 .0206002693428741088... 1
9
8
9 190
csc
dx
x
.02074593652401438021... 7 3
8 322 2
1
64
1
4
32
2
6 21
( ) log( )
x
x xdx
2 .02074593652401438021... 7 3
8 32
1
64
1
4 1
32
2
40
1
( ) log( )
x
xdx
.020747041268399142... B
kk
kk
2
1 2 4( )!
1 .02078004443336310282... 13 3
27
2
81 3
1
54
1
3
1
3 2
32
31
( )
( )( )
kk
Berndt 7.12.3
33 .02078982774748560951... 416
3
3
2
2
0
( !)
( )!
k
k
k
k
.020833333333333333333 = 1
48
1 .020837513725616828...
2 3 11
21
1
2
1
2 5 31
( )k kk
= ( )2 3
21
kk
k
1 .02083953399112117969... 1
2
1
2
1
2
1
4 24 40
cos cosh( )!
k k
k
.020839548935264637084... 1
6 4 2 2 2
4 1
41
cot coth
( )kk
k
... ( )
!
k
kk
1
... 3
22
50125 5
1 25 1
csc cotlog
x
xdx
=
4 2 25 3 5
125 5 5
3
5 2
/
... 197
108
3 3
2
2
5 41
( ) log x dx
x x
... 1
8
1
23
1
2 2
2
20
1
, ,log
x
xdx
=1
2
1
2
1
23 3Li Li
...
9
1
18 6 3
9 3
6
1
18 6 3
9 3
6
i i i i
=1
3 1 31 ( )k 1k
... ( )k
kLi
k kk k
1 15
25
2
1
=691
3276011 ( )
... 1
47
... 1
3
2
3
3
3 1
1
31
21 1
( )
( )
( )
k
k k
kk
k
... ( )
logk
k kkk k
1
4
1
41
1
42 1 k
... 7
1 .02165124753198136641... 41
16 2 10
arctanh1
4
kk k( )
... 1
46
... 42 26 22
3
1
logk Hk
kk
...
k
k
kkk
2
)12(32
1
22
1arcsin22
0
... 7
30720
4 3
51
log x dx
x x
=1
45
... 2
218
3
2
29
16
1
1 3
( )
( )( )k k kk
...
2
54
2
9
1
9 3 3
5 2
2
1
9 3 3
5 2
2
i i i i
1
18 6 3
5 2
2
1
18 6 3
5 2
21 1i i i i
( ) ( )
= 1
11 3 1
3 22 1k
k kk k
( ) ( )
...
12 )32)(22(4
)2(
2
)3(
12
1
kk kk
k
... 4 211
4
1
1 2
1
1
log( )
( )( )(
3)
k
k
k
k k k
... log log
( )
2
3
5
24 1 40
1
x
xdx
=1
44
... 11
42 4
1 1
1 26 42
41
( ) ( )( ) (k k k k k )k k
... 17
16
3
22 4 4
0
4
log sin tan/
x x d
x
... 9
8 12
3 2
4
2
2
1
2
2 2
4 22
log log
( )kk k k
.02291843300216453709... 3
23
13
8
1
9
2 1
932 1
log( )
1
k k
k
kk
k
... 1
1542 kk
1 .0229259639348026338... 1
2 1 244
11 ( )
( )k k
kk k
k
... 50 43
128
28 2
128
71
1536
2 1
2
1
2 61 3
2G k
k kk k
log ( )!!
( )!!,
...
64
2 2 22
1
2 1 22
2 21
sin sec(( ) )kk
... )(,1
2log12
log
2sofzeroa
Edwards 3.8.4
... 31 5
8
7 3
32
7 3
4
2
48 48 42 2
2 4 4 ( ) ( ) ( ) loglog
=H
kk
k ( )2 1 41
... log k
kk2
2 1
.023148148148148148148 = 5
216
1
5
1 3
1
( )k
kk
k
1 .02316291876308017646... 2
2 21
2 3
6
1
12 1
1
12 11
( )
( ) ( )
k kk
=log x dx
x120 1
.02317871150152727739... 233
1920
2
32
1
1643
coth
kk
... 1
43
... 3 2
3148
2
4
1
4 2 1
2
log
( )kk k
k
k
= arcsin logx x
xdx
0
1
.02334259570291055312... 3 2 3 3 2 3 2 31
2
0
( ) ( ) ( )logx x
e edx
x x
.02336435879556467065... 5
36
2
6
log
=( )
( )( )( )( )
1
2 1 2 2 2 4 2 50
k
k k k k k K ex 107b
.02343750000000000000 = 3
128
1
3
3
1
3
51
( ) logk
kk
k x
xdx
... 2
2 4
2
21
sinh
k
kk
.02351246536656649216...
3
5
27
1
4
1
381
4
27
)3(26 )2(3
=
2
3173
2
3134
)3(6
1
381
8
3
)3(233
3 iLii
iLii
i
= dxxx
x
136
2log
... 2 328567
1200072
2
8 71
( )log( )
x
x xdx
=1
42 3 B
... 2
20
6 5112
115
144
1
5
( )
( )
k
k k
dx
x x
...
2
2 211
1
21
1
2 1
cosh( )kk
2
... ( )jkkj
2 114
... 1
162 6k k
s dxk slog
( ( ) )
... ( ) ( )
( ) /
1 2 1
21
1
21 20 1
3k
k kk
kk
k HW Thm. 357
... 16 2 4 2 2 3 41
2 5 41
log ( ) ( ) ( )
k kk
=( )k
kk
4
21
... 21
61
/k
k k
... G k
kk8 64
1
256
1
4
1
6144
1
4
2 1
4 1
22 3
2
40
( ) ( ) ( )
( )
=G
8 64 128
7 3
32
1
6144
1
4
2 33
( ) ( )
=1
41
... 1728
2035)3()5,3(
=
1
0
1
0
1
0
444
1dzdydx
xyz
zyx
3 .02440121749914375973... ( )
!
/k
k
e
kk
k
k
3
0
1
31
.02461433065757607149... 4 3 3
16
2
5 31
( ) log x dx
x x
1 .02464056431481555547... 2 1 272
64 2 12 2 22 32 1
24
23 2
1
( log ) log log ( )( )
H
k kk
k
.02475518879810900399... 61 6
72
1
1 2
2
21
3
k k k kk ( ) ( )( )
7 .02481473104072639316... 5
.0249410205141828796... log( )
5
6 6 62
k
kkk
.02500000000000000000 = 1
40
... ( ) ( ) ( )2 3 43
2
15 4
2
k kk
= ( ) ( )
1 41
1
k
k
k 1
... 1
4 2
=1000
3937 = meters/inch
.02553208190869141210... ( )3 1
8
1
8 113
2
k
kkk k
... 4430
21871
21
8
( )k
kk
k
=1
39
... 23
7
3
2
1 32
1
1
arcsinh
( )k k
kk
k
k
... 2
22 2
18 2 4 2
1
2
1
2 1csc cot
( )
kk
J840
... ( ) ( )
( )6 7 1
71
k
kk
HW Thm. 290
.02584363938422348153... k
k kx d
k
1
162 5
6
log( ) x
.0258653637084026479... 2 5 2 3 4 3 2 42 4
1
( ) ( ) ( ) ( ) ( ) ( )( )
H
kk
k
.02597027551574003216... 1
2 162 1
1
8 1 8 11
( )
( )( )k kk
GR 0.239.7
1 .02617215297703088887... 8
7
8 180
csc
dx
x
.026227956526019053046... 161
900 6 0
1
G
x x dx arccot5 log x
=1
38
.02632650867185557837... 1
2
7
21
2
12
2
0
e
k
kk
k
k
( )( )!
... 1233
2
1
ek
k
k
k
!
.02648051389327864275... log( )
22
3 1 41
5 42
dx
x x
dx
x x
... 3 1
3164 2
1
2 1
G k
k
k
k
( )
( )
... 3 3
4
7
8
1
3 30
( ) ( )
( )
k
k k
=
1
0
1
0
1
0
222
1dzdydx
xyz
zyx
...
1 2
)2(
2csc
2log
kk k
k
=
122
11log
k k
... 4 7 2 5 3 46
1
( ) ( ) ( ) ( ) ( )
H
kk
k
... ( , )
.02681802747491541739...
3
2
6
7
216
1
18
)3(13
381
52 )2()2(
3
=
2
3132
2
314
)3(3
1
2
)3(
381
102 33
3 iLii
iLii
i
=
125
2log
xx
dxx
... 1
81
1
3 13
3
30
1
( ) log
x dx
x
=1
37
... Gk
x dx
x x
k
k
8
9
1
2 5 20
6 41
( )
( )
log
... ( )
( )!cosh
k
k k kk k
1
21
1
2
1
2 2
... 2 4
4164 768
1
2 2 1
k k
kk
( )
( ) J349
... 3
2
5
324 2
0
4
sin tan/
x x dx
...
1 .0275046341122482764... B
kk
k
2
0 2 1( ) !
.02755619219153047054...
( )
( )
log
( )
log5
5 1
1
55 51
p
p
k
kp prime k
3 .0275963897427775429... cosh( )
!
e e
k
k
k2 20
GR 1.411.4
.02768166089965397924... 8
289 40
x x
edxx
sin
... ( )k
kLi
k kk k6
26
1
1 1
... 3
64 2
1
4 3
1
4 1
4 1
3
1
31
( )
( )
( )
( )
k k
k k k J326, J340
.02777777777777777777 = 1
36
1
2 1 2 2 2 4 2 50
( )( )( )(k k k k )k
K ex. 107c
= k
k k k kk ( )( )( )( )
1 2 30 4
... ( )
( )
( ) ( )
( )
1
4 4 1 4 42 1 1
k
kk k kk
k
k
k
k k
1
.02788022955309069406... 115 5
16
1
2 50
( ) ( )
( )
k
k k
... 7 183
2
0
ek
k kk
!( )
... 7 172
4
0
ek
kk
( )!
... 7e
... 1)2(sinh
cscsinlog2
11
1
kk
ke
e k
.02817320866780299106... 3 3
128 1
2
51
2
40
( ) log
x dx
x x
x dx
e x
... 1
...
8csc
8
3sinlog222log8
8
7cot
16
1
=
21log222log8
8
7cot
16
1
=
12
2 864
1
8
)(
kkk kk
k
.02839399521893587901...
145
21
2
cos
1cos
)1(cos1
51cos4
1cos21
kk
k GR 1.447.1
... 3
2
4 1
3
1
1
e
k
k
k
k
( )
( )!
=1
35
...
( )log
55
1
k
kk
... ( )ck
kLi
kkcc
kc
1 12
122
21
...
1
31
2)!1(
)1(
4
132
kk
k
k
k
e
... 3 3
22
4 1
2 3
41
( )log
log
( )
x
xdx
... log log11
30
12
x
x dx
= 122
34 2
1
6
7
6
2
3
1
72
2
3
7
61 1 2 2
log ( ) ( ) ( ) ( )
.0290373315797836370... 1
1000
9
10
2
5 12 2
2
51
( ) ( ) log
x dx
x
.0290888208665721596...
108 92 20
dx
x( )
3 .02916182001228824928... sinh
( )
3
7 31
3
2 21
kk
.02929178853562162658... 1
48
1
16
1 2
420
e k
k k
k
( )
( )!
1
... 4 1 2
2
k
k
k( ( ) )
... 1
34
.029524682084021688... c34 2 21
2412 2 8 3 3 2 6 4 ( ) ( ) ( ) ( )
Patterson ex. A4.2
.0295255064552159253... 7 3
4
56
272
1
2 1 32
( )
( )
kk
= log2
6 41
x
x xdx
= x x
xdx
4 2
20
1
1
log
GR 4.261.13
.02956409407168559657... 1
4
1
36
5
6
1
6
1
3 21 1
1
21
( ) ( ) ( )
( )
k
k k
= 1
36
5
6
4
31 1
6 31
( ) ( ) log
x dx
x x
...
0 2)12(22
1
8
1,
2
3,
2
3,1,1,
2
1
k k kk
kHypPFQ
...
12 )32)(12(4
)2(
8
)3(9
6
1
kk kk
k
1 .02967959373171800359...
3 2
40
1
1
4
163
41
/
log
x dx
x
.029700538379251529... 2
3
9
8
2 1
2 42
( )
( )!
k
k kk
!!
...
3 33
3 3 1330
log
x dx
e x GR 3.456.1
=
log x dx
x1 330
1
GR 4.244.2
.0299011002723396547...
2
214
39
16
1
1 2
k k kk ( )( )
J241, K ex. 111
... 3 2
2
2
2
5
4 2 1 2
2
1
log log
( )( )(
3)
H
k k kk
kk
.0300344524524566209... 2
27
3
21
2
1
log ( )
k kk
k
H k
2
... log ( )( )
6
5 51
52
kk
k
k
k
1 .0300784692787049755... ee erf k
k k kk
3
20 1
1
21
2
2 1 2 11
, ,( )!!
( )!! !(
)
2 .0300784692787049755... e erf k
k
k
k
1
2
4
2 10
!
( )!
.03019091763558444752... ( )31
2
1
21 1 i i
= )2()2(2
1)3( ii
=
1
1
235
1)32()1(1
k
k
k
kkk
... 3 2
4 401024
x dx
e ex x
.03030303030303030303 = 1
33
1 .03033448842250938749... 6 3
7
( )
... e
e
...
( )( )
3
42
2 2 c Berndt 9.27
1 .03055941076... j5 J314
.03060200404263650785... 1
64
1
4
1
16
1
4
1
2 16
1 2
1
cos sin( )
( )!
k
kk
k
k
2 .03068839422549073862...
2 2
322
1 2 35 5
1
2
( )( )
k k
k kk
= ( ) ( )( ( ) )
1 12
2
k 1k
k k
.030690333275592300473...
12
11
1
2
log ( )( )
kk
k
k
k
... dxee
xxx
022
34
18
3
240
.03095238095238095238 = 13
420
1
2 1 2 5 2 90
( )( )(k k k )k
.03105385374063061952... 132
1
2 1
3 1
31
( )
( )
k
k k
.03111190295912383842... 33
48 48
9 3
24
1
2
2
31
( ) ( )
( )
k
k k k
... 15 5
8
45 3
2140 2 126
1
1
1
5 51
( ) ( )log
( )
( )
k
k k k
... log
( )( )
( )(
2
4
1
61
1
1 22
k
k k kk )
5 .03122263757840562919... 2 2
2 1
1 22 1
1 1
k
k k
k
k k
k
( )
( )!sinh
... 2 1
216
2 2 31
1 2
log( )
( )( )
k
k
k
k k
... 1
32 84
1
16
1
21
csch
( )k
k k
.03125002980232238769... 1
2
5
32 151 1
k
kk
k
k
( )
.03125655699572841608... 23
2 2 81
2
22
3
log Li
1 .03137872644997944832... 1
51 ( !)kk
.03138298351276753177... 12635
3 9
1
1
2 4
5 51
k kk ( )
1 .03141309987957317616... cosh( )!
1
4
1
2 160
k kk
... 1
642
5
82
9
8
1
64
5
8
9
81 1 , , ( ) ( )
= ( )
( )
1
4 1
1
21
k
k k
= log x dx
x x6 21
... 12
2
1
11
12 2
2
logsinh
coth log
k kk
= ( ) ( )
11
2 11
k
k
k
kk
.032067910200377349495... 1
6 64
5 2
12 1 1
2
41
log log
( ) ( )
x
x xdx
... 527
7350
2 2
35
1
2 7
1
1
log ( )
( )( )
k
k k k k
=1
32
2
51
log x
xdx
... 123
2
29
6
1
2 2 6
1
0
log( )
( )
k
kk
k
k
... 1
3
=1
31
... )6(48
203)3(3)1,2,3( 2 MHS
... ( ) ( ) ( ) ( )5 3 3 2 4 5 2 15
=1
12 51 k kk ( )
... 5
2 4
1
1 2
2
2 21
k k kk ( ) ( )
... log
( )
2
4 2 2
2 1
8 2 102
G k
k kkk Berndt 9.6.13
1 .0326605627353725192...
15
155 )13(
1
)13(
11
243
)3(242
kk kkG
J309
1 .03268544015795488313... 1
31 k k
k
.03269207045110531343... 3
4
5
6
1 2 1
2 32
( ) ( )!
( )!
k
kk
k
k
!
... 96 646
0
dx
x
.03283982870224045242... 19
72
2
3
1
2 4
1
1
log ( )
( )( )
kk
k
H
k k
... 2
61300
log x dx
x x
.03302166401177456807... J
k kk
k
k1
0
4
41
4
1
( )!( )!
... 365
48
1
4 1
1
0
log( )
( )(
3)
k
kk
k
k k
... 7 3
16 192 2 1
4
41
( )
( )
k
kk
... 2
61
k
k k k!
...
3
3
2 6
1
61 3
3 3
21 3
3 3
2
2
( ) ( )i
ii
i
= ( ) ( )
1 3 2 111
15 2
2
k
k k
kk k
.0333333 = 1
30 2 4 B B
1 .03348486773424213825... 1
41 k kk !
... 3
30
.03365098983432689468... 1
27
1
3
13 3
81
2 3
7291
3
( ) ( )
= 1
1626
1
3
1
3 31 2
31
( ) ( )
( )
k
kk 1
.033749911073187769232...
1
02
6
)1(
)1log(2log3
34
5dx
x
x
.03375804113767116028... 5 2 3 4 51
1 51
( ) ( ) ( ) ( )( )k kk
= 1
6 52 k kk
.03388563588508097179... ( ) ( )k k
kk
0
1 2 1
1 .033885641474380667... pq k
kk
( )
21
.03397268964697463845... 3
2 12
2
2
2
2
1
2
2 2
3
log logLi
.03398554396910802706... ( )
( )!sinh
2 1 1
2 1
1 1
1 2
k
k kk k
k
.03399571980756843415... J Fk k
k
k4 0 1
0
21
245 1
1
4( ) , ,
( )
!( )!
LY 6.117
.03401821139589475518... 1
144
5
12
11
12 11 1
61
( ) ( ) log
x dx
x
.03407359027997265471... log
sin tan/2
2
5
164
0
4
x x d
x
1 .03421011232779908843... HypPFQk
k kkk
1
2 1 12 2
1
4
2 1
16 102, ,
, , ,( )
... 3 3
22 2 3
1
36 31
loglog
k kk
J376
... 1
2
11 1
12
1
ee
erf erfix
dx
sinh
... 4
2 34
2 2 11
3
4
2
20
1log( !)
( )!( )
k
k k
k
kk Berndt Ch. 9
... 7
6
7 170
csc
dx
x
... 1
29
... 1 1
3
1
3
1
1e k
k
k
( )
( )!
... 12 2 21
1 340
log( )(
)
k kk
.034629994934526885352... 3
2 122 2
3 3
4
12
4 32
log
( ) ( )k
k k k
... 1
1024
7
8
3
8
1
4 12 2
1
31
( ) ( ) ( )
( )
k
k k
... sin
arctan
x
xdx
0
1
2 .03474083500942906359...
22
3coscosh)(sinh
= sinhcosh cos cosh cos
4
1 1 3 1 1 33
= 11
61
kk
Berndt 4.15.1
1 .03476368168463671401... 1 11 1
1
2 12
2
2
e
k k e
k
kk
k
kk
k
k
/
/ ( )!
( )
=
Re( ) 2 1
1
k
ikk
2 .03476368168463671401... 1 11
11
1
2 11 2
e
k k e
k
kk
k
kk
k
k
/
/ ( )( )!
( )
.03492413042327437913... Ai ( )2
... 21
2
1
2 1 24 40
4
1
Lik
Hk
k
k
kk
( )
( )
...
1
2
1
4
1
2
1
2
1
2
1
2
i i i i
= ( ) ( )
1 4 1 111
15
2
k
k k
kk k
.035083836969886080624... 1
21 1
2
log( )
k
kkk
... ( ) ( )( ) ( )
( ) ( )( )
( )2 34
4
3 6
42 3
3
22 5
2
=H
k kk
k ( )
1 51
... 2 3 15
12 csc
AS 4.3.46, CGF D1
.03532486246595754316... 11
32
1
1
2
2 32
( )
( )
k
k k
... 1
9
...
3
4
1
2 2 1 2 2
1
1
( )
( )( )(
k
)k k k k J244, K ex. 109c
...
1
125
4
52 ( )
.03561265957073754087... ( )
( )
5 1
5
...
1
12
)12()1(
2
1
kk
k kii
=1
28
... 1
55
1p
k
kk
p prime k
( )log ( )
... 2
25 5
1 2 1 22
21
3
21
( ) ( )!
( !)
( ) ( )!
( !)
k
k
k
k
k k
k
k k
k
1 .03586852496020436212...
1
4)14()4()14()24(k
kkkkk
... 1
11 55
2
1
1kk
k
k
k
( ) ( ) 1
... 2 4 2
4132 384
7 3
16 2 1
( )
( )
k
kk
.03619529870877200907... 2 7 2
81
1
12 31
log
log( )x
dx
x
.03635239099919388379... 1
2
1
2
1
2 2 12 11
arctan( )
( )
k
kk k
... k
kk
k
k
k6
2
1
111 6 1
( ) ( ) 1
... 1
5
6
5
1
5 611
log( )
H nk
kkn
... 2 2 23
2
1
2
log
... log( )
log4
5 5 51
1
5
1
52 1
k
k kkk k k
1 .03662536367637941437... 1
36
1
6
1
6 5 11
21
60
1
( )
( )
log
k
x dx
xk
... 2 14
2 1
0e k
k k2
k
( )
!
...
1
1
27
2
1)27()1(1 k
k
k
kk
k
... 3 1
2 31128
3
32
1
2
1
4 1
( )
( )
k
k k
...
032 )4(256
3
x
dx
...
12 )22)(12(4
)2(
4
)3(7
4
1
kk kk
k
... dxxx
li log2sin1
2log225
1 1
0
... ( )5 ... 6
= 1
27
...
1235
1)38(1
kk
kkk
.037087842300593465162... 1
3
3
27 131 1
k
kk
k
k
( )
...
1225
1)27(1
kk
kkk
... ( )
log( )6 1 1
11
6
2
k
kk k
k k
2 .03718327157626029784...
21
3
5
32
...
03
)2(
)316(
1
16
3
8192
1
k k
... 9
4
3
2
7
8
1
2 122
log( )
( )
k
kk k k
... coth22 2
2 2
e e
e e
=1
2
16
2 20
k
kk k
B( )!
AS 4.5.67
.03736229369893631474... 1
2
3
64
1
4 1
2
2 31
( )kk
J373, K ex. 109d
... 1 2 4 32 4
1
( ) ( )( )
k
kk
...
13
)2(3
)912(
1
4
1
3456
1
432
)3(7
1728 k k
...
1215
1)16(1
kk
kkk
...
21
1!
1)5( 5
k
k
k
ek
k
... ( )
log( )5 1
11
5
2
k
kk
k k
... 3 2
4
19
20 8
1
4 4 11
1
42
1
2
log
( )( )
( )
k k
k
k
kk
k
.03756427822373732142... dxe
xdx
xx
xx
04
2
15
2
1
log
32
)3(
.03778748675487953855...
032 )3(348 x
dx
... 6
...
13
)2(
)69(
1
3
1
1458
1
k k
... 16 29
3
1
4 20e k
k
kk
( )
( )!
... 3 3
42 2
9
4
15 3
2
( )log
( )
k
k k k
=
13
1
)2()1(
)1(
k
k
kkk
.037901879626703127055... 1
2
2 2
3
1
27 331
log ( )k
k k k [Ramanujan] Berndt Ch. 2
...
12
51)5(
1
1
kk
kk
... ( ) log( )4 1 1 1
1
4
2
k
k
k
kk k
...
1
2)12(
812cos
7
1
k k
...
13
)2(
)58(
1
8
3
1024
1
k k
...
0
sinlog4
2log
22
1dxxxex x
1 .0381450173309993166... k
kk
5
52 1
.0381763098076249602... 80 4 48 2 6 31
2 12
3 31
log ( )( )k kk
.03835660243000683632... dxx
x
1
02)22(
)1log(
8
2log1 GR 4.201.14
1 .03837874511212489202...
0
1
1112
)1(
k
k
k
.0384615384615384615 =
0
5cos26
1dxex x
.0385018768656984607... cossin
cos sin cos22
22 1 1
=
1
21
2/32/1 )!2(
4)1()2()2(
k
kk
k
kJJ
31 .03852821473301966466... 3 3
.0386078324507664303...
022 )1)(1(2
1
2
1xex
dxx Andrews p. 87
= 1log4
log1
022 x
dx
x
x
GR 4.282.1
.03864407768699789287... dxxx
x
14
2
)1(
log
36
331
... dxxx 22/
0
23
sin488
... 7
8 162
21 3
2
64 20 8 1
2 2 1
2 4 3 2
42
log( )
( )
k k k
k kk
=k k
kk
3
2 2
( )
...
122
21
02
22
)1(
log
)1(
log)3(3)2(2
xx
dxx
x
dxxx
... 3
)3(
... 1log16
112log2
28
12
1
022
x
dx
x GR 4.282.9
.03886699365871711031... 21
2
1
2
5
41
1
2 2
1
2 11
arctan log( )
( )
1
k
kk k k
... 7 3
216
1
6 3 21
( )
( )
kk
=3929
100800
1
5 81
k k kk ( )( )
... 3
11
!
1)23(
22
1
k
kk
ekk
k
... 1
83 4 2 2 ( ) ( )i i
= 5
8 2
1
42 2
15
2
( ) ( )i i
k kk
= ( )4 1 11
kk
... ( ) log( )3 2 1 1
1
3
22
k
k
k
kk k
... 116
21
163
2 3
2
31
( )( )
k
kk
... 25 5
225 2
11
4
1
4 1
1
1
loglog
( )
( )(
2)
k
kk k k k
... 1
12 32 ( )kk
.03945177035019706261... 1
0
322
log)1log(54
12dxxxx
... 1)2(1
1
4
9log2
2
kk
k
k
1 .03952133779748813524... 2 1cos e ei i
3 .03963550927013314332... 30 5
22
( )
1 .03967414409134496043... 1cschcoth2
1 22
=
2222
1
1
1
212(2)1(
kk
k
kkkk
1 .03972077083991796413... 3 2
23
1
9 2 10
log
( )
arctanh1
3 kk k
= 1 21
64 43 21
k kk
[Ramanujan] Berndt Ch. 2
...
02
)2(
)35(
1
5
3
250
1
k k
...
1
12
1)3()1(4
13
3 k
k kk
... dxx
xx
1
0
22
1
log)1(
4
5
3
.04000000000000000000 = 1
25
.0400198661225572484... 216
251)3()4,3(
=
1
0
1
0
1
0
333
1dzdydx
xyz
zyx
.04024715402277316314... ( )2 1
21
kk
k
1 .04034765040881319401...
3 1041 2 / ( )
4 .04050965431073002425... 7
.040536897271519737829...
15
2
)1()1,5(
2
)3(
4
)6(3
k
k
k
HMHS
.040634251634609178526... 13
30 8
2 1
161
( )kk
k
... sin cos( )
( )!
1
2
1
2
1
2
1
2 1 41
k
kk
k
k
... 23 1 5 1
161
2 1
4
1
sin cos( )
( )
!
k
k
k
k
... ( )( )
3 12 2
22
f k
kk
Titchmarsh 1.2.14
= 1
322 ( )jkkk
.04084802658776967803... 13
2
3
4
2
3
1
6
1
6
1
621 2
log log ( ) ( )
k k
k
k
k
kk
... 1681
292
1
2
383
2
1
2 4 2
1
1
cos sin( )
( )! ( )
k
kk k k
... ii eLieLi 2
12
1
...
54 2 8140
dx
x
.04132352701659985... B
kk
kk
2
1 2 2( )!
.04142383216636282681... 1
2
1
2 21 1
1
tanh ( ) k k
k
e J944
.04142682200263780962... 7 3
16 64
1
128
3
4
1
4 1
32
31
( )
( )( )
kk
1 .04145958644193544755...
3
1
3
12
2
3log
6 22
32
LiLi
.0414685293809035563... 11
3
2
3
3
2 3
1
3 1
1
1
e
e
k
k
k
cos( )
( )
!
.041518439084820200289... 112 4 2 2
12
4 22
tanh coth
( )k
k k k
.04156008886672620565... 26 3
27
4
81 32
1
27
4
3
32
2
5 21
( ) log( )
x
x xdx
2 .04156008886672620565... 26 3
27
4
81 3
1
27
1
3 1
32
2
30
1
( ) log( )
x
xdx
= x dx
e ex x
2
20
.04156927549039744949... 13
21
6 2 3
3 3
2
1
6 2 3
3 3
2
( )
i i i i
= 1
5 22 k kk
.04160256000655360000... 1
5
2
5 121
21
k
kk
k
k
( )
.04161706975489941139... 1
4 2
1
2
1
2
1
2
1
2
1
4
1
1
cosh sin cos sinh( )
( )!
k
k
k
k
.04164186716699298379... cos cosh( )
( )!
1
2
1
21
1
4
1
1
k
k k GR 1.413.2
2 .04166194600313495569... k
k Hkk !
1
.04166666666666666666 = 1
24
1
3 1 3 4 3 70
( )( )( )k k kk
C132
1 .04166666666666666666 = 25
24
1
221
k kk ( )
1 .04166942239863686003... 12
1 ( )!kk
1 .04169147034169174794... 1
21 1
1
2 4
1
4
1
40
cos coshcos
( )!
e
e kk
.04169422398598624210... ( )
! (
k
k kkj
jk )!
2 22
1
... 1
42
2
16
5 2 2
5 2 2
3
8arctan log
log
( )
arctan arctan2 1
8
2
81 2 1 2 4 4
2
dx
x x
.04171627610448811878... e
e
k
kk16
1
16
1
8 41
sin
( )!
1 .0418653550989098463... e
e kk3
2
3 3
3
2
1
3 10
cos( )
! J804
.04188573804681767961... 1
6
9
7 4 22
log
dx
x x
... ( )
log
15
2
k
k k k
721 .04189518147832777266... 3
4
6
...
1
22 2 1 2
1
2 1
log cos log sincosk
kk
GR 1.441.2
.0421357083215798130... 4 2 29
4
1
2 32
1
1
log log( )
( )(
)
k
k
k
H
k k
1 .0421906109874947232... 21
22
1
2 1 41 2 1 2
0
sinh ( )( )!
/ /
e ek k
k
= 11
4 2 21
kk
GR 1.431.2
.04221270263816624726...
1
2 4 2 2 2
4 1
24 4 41
cot coth
( )kk
k
1 .04221270263816624726... ( )4
2
1
2 114
1
k
kkk k
=
444 2
coth2
cot242
1
1 .042226738809353184449...
1)1(
122
22
12
2
1
12
1
kkkk
kkk
[Ramanujan] Berndt IV, p. 404
... 16 21
1 4
1
1 42
1
2
1
21
( )
( / )
( )
( / )
k k
k k k
.0422873220524388879... B k
kk
k
k
2 2
0 !
.04238710124041160909... log
( )( )( )(
2
4 24
1
4 3 4 1 4 1 4 31
)
k k k kk
J242
.04252787061758480159... 1 1
21
11
12
= 1
1
2 12 2
12 1
1k k
k
kk
k
( )
... 8
125
12
125
4
31
41
2
1
log ( )k kk
k
k H
.042704958078479727... 64
2 27 3
4
1
2 1
2
31
log
( )
( )k kk
... 1
2
1
2
1
2 2 11
cosh sinh( )!
4
k
k kk
... arctan log2
4 4
3
4 142
dx
x
.0428984325630382984...
2 2
31
102 2 3 4
( ) ( )
1 .042914821466744092886... 1
2
1
2 1 2csc
e dx
e
x
x Marsden Ex. 4.9
.04299728528618566720... 163
2
58
9
1
2 2
1 2
1
log( )
( )
k
kk
k
k
... dxx
x
1
06
3
1
)1(log
32
32log
2
32log53
.0431635415191573980... 1
6 12 3
3
4
1
3 2 3 1 3 3 11
log
( )( ) (k k k k )k
GR 0.238.3, J252
5 .04316564336002865131... e
.0432018813143121202... ( )k
kLi
k kk k
1 14
24
2
1
... 1
2
2 1
2 1
4
4
Mel1 4.10.7
.04321391826377225... i i ii i2 1 2 2 i ( ) cos( log ) sin( log )
.0432139182642977983... ( ) //
/ /( )2 1 1 4
2
1 4
11 4 3 42 1
1
2
e k
k
J114
.04321740560665400729... e k
k
2
1
.04324084828357017786... arctanh1 1
2 2
1
2 12 10e e k kk
log tanh( )( )
1 .04331579784170500416... ( !)
( )
k
k !k
3
0 3 1
1 .04338143704975653187... ( )3 1
2
1
214
1
k
k kkk k
.04341079156485192712... 23
4
13
22
5 2
2 2 1 2
2
1
3
loglog
( ( )(
kH
k k kk
k )k
.043441426526436153273... 2
3
2
21
1
2 2 1 2 31
arcsinhk
k k k( )( )
... ( ) ( ) ( )
1 4 4 111
15
2
k
k k
k kk
k k
= 1
23
... ( )k
kLi
k kkk k
1
2
2 22
22
2
.043604011980378986376...
13 )12(3)12(27
1
3
2log
4
3log
k kk
[Ramanujan] Berndt Ch. 2
... arctan( )ee ex x 1
1
2 20
1
...
( )
( )
( )4
5 51
k
kk
Titchmarsh 1.2.13
.04382797038790149392... 3 2
4 8
1
12
1
4 4 1
1
42 2
log
( )
( )
k k
k
kk
k
1 .04386818141942574959... 16
15
1
4 2 1
2
0
arcsin( !)
( )!
4
k
k kk
... 23
6 2
8
33 2
0
4
sin tan/
x x d
x
... 7 2161
1 6
1
1 63
1
3
1
31
( )
( / )
( )
( / )
k k
k k k
... ( )
( )! (
1
1 2
1
1
k
k k k ) Titchmarsh 14.32.1
... 1
22
1
120
csch( )k
k k
... ee
ke
k
k
!0
... 1
16 2
1
4
21 2
1
( )k
kk
k k
k
...
245
245
1)4()3()2(2log
3
14
xx
dx
kkk
.04427782599695232811... 2 2 2 32
34 2
2 1
16 2 11
log log( )
k
k k kkk
... 1
4 48
1
2 2
2 1
21
5 31
( )
( )
k
k k
dx
x x
... 22
1 1( ) ( )k kk
.044444444444444444444 = 2
45 1 41
arccosh x
x( )
.0445206260429479365... ( )
( )
3
27
1
3 31
kk
... 3
32
1
4 12 31
dx
x( )
... 9 29 3
2
5
4
1
4 11
loglog
( )(
2)
kk k k k
...
1 5
1
5
1
kkLi
... 3
8 2
1
4
1
4 2 1
21 2
1
arcsinh ( )
( )
k
kk
k
k
k
k
... ( )
logk
k kkk k
1
2
11
1
21 1 k
... 13
... 144391
2
53
2 2 11
3
e
e k
k kk ( )!( )
1 .0451009147609597957... 2 21
8 2 10
arctanh1
2 2
kk k( )
... li( )2
... 8 27
2 22
1
log
Hkk
k
... 1 21
2
1
2
5
4
1
2 22 11
1
arctanh log( )k
k k k
=1217
26880
1
4 81
k k kk ( )( )
... ( ) ( )4 5
... 1
2 2
13 22
0
coth( )
/
kk
... 2 2
3
5
12
2 4
22
log ( )!
( )!
k
kk
=1
22
... 1
7
... 7 4
8
3 3
4
1
1
1
41
( ) ( ) ( )
( )
k
k
k
k
.0455584583201640717... 17
83 2
1 31
log( )
dx
e ex x
... 2
21216
2
36
1
6
( )
( )kk
... 32 2
1
16
1
16 22 1 log
=log
log
x
x
dx
x 2 20
1
216 1 GR 4.282.10
.04596902017805633200...
9 18 3
3
6
3 3
18 3
7 3
6
3 3
18 3
7 3
6
log i i i i
=1
27 131 kk
... 10
3
3
2
47
36
1
3 11
log( )(
3)
kk k k k
... log ( )( )
5
4 41
42
kk
k
k
k Dingle 3.37
= ( )( )
log
14
1
41
1
42 1
kk
k k
k
k k
k
... Ik kk
31
21
6
1
3( )
!( )!
1 .0460779964620999381... 1 22 2
1
2 4 21
( ) cot
k kk
=( )2 2
21
kk
k
.0461254914187515001... 8
2
2
1
4 2 4 1 41
log
( )( )k k kk
J251
= 1
4 1 211 ( )k j
kj 1
J1120
1 .046250624110635744578... F kkk
2 12
2 1
( )
.046413566780796640461... 5
6
4 2
3 6
3 3
4
5 3
4
14
2
logcot cot
( ) i i
k k
k
k
2 .04646050781571420028... 1
3 130
kk
.04655015894144553735... 1
2
3
12
1
6 1 6 11
( )(k k )k
.04659629166272193867...
2 1
1 2
1
2 21
2
1
4
1
2 2F
k
k
k
k
, , ,( ) ( !)
( )!
...
2 122
22 2
0
2
log logsin
/
x dx GR 4.224.7
= log2
20
1
1
x dx
x GR 4.261.9
.04667385288586553755... 11
41
2
1 2
0e
k
k k
k
k
( )
!( )
.04672871759112934131... 5
18
2
3
1
3 9
1
1
log ( )k
k k
... csc ( ) csc ( )/ /1 11 4 3 4
... H
kk
k
2
51 1( )
...
22
)(
kkk
k
1 .046854285756527732494... 1
1
1
2 22k k k k kkm
km( ) log log
... ( )
( )! !
k
k k kI
kk k
2
1 2
2 12
...
21 2
1
2
1
2 1
3
20
4
arctansin
cos
/ x
xdx
... sin
/
3
3
0
3
... 5 5log
.04719755119659774615... 3
12 1
2 4 2 1
2 1
16 2 11 1
( )!!
( )!! ( ) ( )
k
k k
k
k kkk k
k
... 3
1
21
1
6 5
1
6 11
1
arccos ( )k
k k k J328
= 2 1
16 2 10
k
k kkk
( )
= dx
x
x dx
x
x dx
x60
4
60
1 2
301 1
/
1
... 1
4 ( ) ( )i i
... 4
5193
30 4
31
( ) ( )a k
kk
Titchmarsh 1.2.13
93648 .04747608302097371669... 10 ... 2
...
2 4 4
413
7
909 3
1
( )log
( )
x
xdx
=1
21
... 5
43
1 1
3 25 32
31
( )( ) (k k k k k )k k
... log ( ) log( )
2 2 11
11 2
2 22 2
k k
k
kk k
1
... log
( )( ) (
2
4
1
8
1
4 2 4 4
1
2 112
2
)
k k kkk
k k
... 2 2
118
1
2 2
2
( !)
( )
k
k !k
...
1
5
1
21
21
2
1
4 32
i i
k kk
= ( )( )
12 1
41
1
kk
1
k
k
1 .0483657685257442981... 1
2 1 233
11 ( )
( )k k
kk k
k
... 1
36
2
3
7
6
1
3 41 1
20
( ) ( ) ( )
( )
k
k k
=log x dx
x x5 21
... ( ) ( ) ( ) ( )( )
3 3 5 3 4 61
3
61
k
kk
... 24 2 3 41
1081
22
0
1loglog( ) log
x x x dx
... 3 1 1
161
2 11
4
1
cos sin( )
( )
!
k
k
k
k
.04878973224511449673... 2 32035
864
2
6 51
( )log
x
x xdx
1 .04884368944513054097... ( ) ( ) tanh2 3
3
3
2
15 4
1
3
k k kk
222 .04891426023005682... 8
364 2
33 2
22
0
log ( )/x Li x dx
.04904787316741500727... ( ) ( )( )( )
2 2 3 41 2 3
1
k
k kk
= ( ) ( )
1 21
1
2k
k
k k 1
1 .04904787316741500727... ( ) ( )( )
2 2 3 39 11
1
2
2 32
4
k k
k kk
= ( ) ( )
1 11 2
3
k
k
k k
3 .04904787316741500727... ( ) ( )2 2 3 1 12
1
H
kk
k
4 .04904787316741500727... ( ) ( )( )
( )2 2 31
11 13
2
2
1
k
kk k
k k
.04908738521234051935... 64 4
2
4 20
x dx
x( )
5 .049208891519142449724... 49
16 2
4
0
e k
k kk
!
.0493061443340548457... log ( )3
2
1
2
2 1
3
1
27 32 11
31
k
k kkk k
= dx
x x4 22
.049428148719873179229...
41 1 1
1
11 4 1 4 3 4
42
( ) csc ( ) csc ( )( )/ / /
ik
k
k
... 1
87 6 3 3 27 3 12
1
32 2
log log ( )1
=H
k kk
k ( /
1 31 )
.04952247152318661102... 263
315 4
1
2 9
1
1
( )k
k k
.04966396370971660391... sech 3
2
13 3
e e
... 9
128 2
1
4 2 1
24
1
( )
( )
k
kk
k
k
k
k
...
1
1
3
1
3
1)1(
kk
k Li
.04974894456184419506...
1
02
4
)1(
)1log(2log2
42dx
x
x
... ( ) ( !)
( )!( )
1
2 1 2 1
1 2
1
k
k
k
k k
... 13e
... 1423 1
2 2
1 2
1
e
k
k
k
kk
( )
( )!
... 5 2
6
19
36
1
2 21
log
( )(
3)
kk k k k
... 115
2 5k kx dx
k xlog( )
... 2 1
2112
2 4 21
1 2
log( )
( ) (
k
)k
k
k k
.05000000000000000000 =1
201
92
4
kk
= k k
kk
( )
( )
6
3 21
J1061
1 .05007513580866397878... 2 31 2 1 4 1 2/ / / CFG F5
.05008570429831642856... ( )3
24
.05023803170814624818... 51
48 24
3
2
1
2
2
31
( )
( )k kk
.05024420891858541301... ( ) log
( ) ( )
( )2 2
3 3
4
1
1
1 2
31
k
k
k
k
2 .05030233543049796872... 12
log ( ) kk
.050375577025453... ( ) log
1 2
22
k
k
k
k
.05040224019441599976...
6
1
21 3
1 3
41 3
1 3
4
i
ii
i
= 1
2 1 31 ( )kk
1
.0504298428984897677... 8 22 4
3 2 12
4
0
1
G x
log log( ) log x dx
.05051422578989857135... ( ) log
( )
3 1
4 1
2
5 31
2
2 20
x
x xdx
x dx
e ex x
.05066059182116888572... 1
2 2
1 .05069508821695116493... 1
25
1
5
1
5 41
21
( )
( )
kk
= 2
125
1
16
1
25
9
5
21
( )
.05072856997918023824... Ik kk
40
21
4( )
!( )!
LY 6.114
2 .05095654798327076040... 2
21
1
2 21
sinh
kk
1 .05108978836728755075... cosh( )!
/ /1
2
1
2
1 1
20
e e
k kk
AS 4.5.63
2 .05120180057137778842... 21
51
/k
k k
.05138888888888888888 = 37
7201
12
113
logx
dx
x
.05140418958900707614... 2
51192
log x dx
x x
.0514373600236584036... 2
3 2
16 16
1
82 2
0
2loglog
x e x dxx GR 4.355.1
.05145657961424741951... 244
3
9
4
1
3 1
1
1
log( )
( )(
2)
k
kk k k k
1 .05146222423826721205... 22
5 5
3
5 csc
.05153205015748987317... ( ) ( ) ( )( )
2 6 3 6 47
8
4 1
1
2
42
k k
kk
= Berndt 5.8.5 ( ) ( )
1 11 3
1
k
k
k k 1
1 .0515363785357774114... H kk
k
( ) ( ( ) )3
1
1 1
... 504013700
17
1
1
e
k
k kk
k
( )!( )
2 .05165596774770009480... 1
2 2
2 2
24 2 121 1k k
k k
kk
k
( )
1 .05179979026464499973... 7 3
83
1
2 1 230
2
1
( )( )
( )
k
H
kk
kk
k
.05181848773650995348... 178
225
1
5
2 1
2
1
2 51
2
( )!!
( )!!
k
k kk
J385
.05208333333333333333 =5
96
1
2 4
1
1
( )
( )( )
k
k k k k
4 .05224025292035358157...
3 2
032
1
x x d
x
sin
cos
2 x
log log ( )3
4
2
3 4 3
1
36 6 621 2
k k
k
kk
k
.05225229129512139667...
dx
x x x x x x6 5 4 3 21
=
40
2
31
3
6
( )
( )
( )
k
kk
2 .05226141406559553132... Titchmarsh 1.7.10
ek
kk
cos sin(sin )sin
!3
1
32
.05226342542957716435...
( )
( )!( )
1
2 1 2 1
1
1
k
k
k
k k .05230604277964325414...
( ) !
( )!( )
1
2 1 2 1
1
1
k
k
k
k k .05238520628289183021...
1
1
1 1
3 33 31 1e e kk kk k
cosh( ) sinh( ) .05239569649125595197...
5 6 2
16 1 32
log
logx
x
dx
x .05256980729002050897...
k jk
k kk
jkkj
kk2 1 2 2 22
1 11
0
1( )
( )
k k
4 .0525776663698564658...
1
19 .052631578947368421 =
1
2
10
9
1
19
1
19 2 12 10
log( )
arctanhk
k k K148 .05268025782891315061...
.05274561989207116079... 8
1
6
2
4
1
4 2 4 31
log
( )(k k )k
3
3
12
2
3
2 1
41
log log ( )kk
k
.05290718550287165801...
.05296102778655728550... 2 24
3
1
4
2 1
432 1
log( )
1
k k
k
kk
k
17
720
1
2
4
40
( )
( )
k
k k .05296717050275408242...
3 2
2 2132 1
Gx
xdx
log
( ) .05298055208215036543...
1 .053029287545514884562... 2
22 2
164 8
1
8 1
1
8 7csc
( ) ( )
k kk
1
6 .05305164769729844526...
2 3 22
3
2 2
2 1 31
arcsin( )!!
( )!!
k
k
k
kk
6 .0530651531362397781...
3 3
2
7
42
1
1 32
( ) ( )
( )
k
k k .0530853547393914281...
log log2
4 31
2 2
0
1
1
x dx
x x
x x
x
dx
=
( )jkkj
113
1 ...
8 2 2 2 31
2
3
24 31 1
log ( ) ( )( )
k k
k
kk
k
1 .053252407623515317...
.05330017034343276914...
4
37
12
31
4
37
12
31
3
2log
6
iiii
=1
2 1 31 ( )k 1k
... 2 2 22
2
2 11
log log log csc( )
k
kkk
=
2 11
2 21
logkk
.053500905006153397... 1
40 ( )!k k!k
1
100
2
5
9
10 11 1
51
( ) ( ) log
x
xdx .05352816631052393921...
.05357600055545395029... ( )jkkj
5 111
.05358069953485240581... 1
31 ( )!k k
!k
( )k
kk
12
32
.05362304129893233991...
k
kk
!!
( ) !
40
.05362963066204526106...
1
11 Fkk
2 .05363698714066007036...
1 10
1
4
5
4
1
4
1
4 4 1F
k kkk
, ,! ( )
1 .0536825183947192278...
2
32
7 3
16
1
27 2 1
( )
( )
k
kk
.05391334289872537678...
43867
1436417 ( ) 3 .05395433027011974380 =
log log
( )
3
8
1
12 2 31
x
xdx .0539932027501803781...
3
41
3
2 31
logkHk
kk
1 .05409883108112328648...
2
2
11( )
( )k
k
k 1 .05418751850940907596...
.054275793650793650 =5471
100800
1
3 81
k k kk ( )( )
11
1
11e e
dx
e ex x
log( )
.054617753653219488...
33
2 21
12
7
6
2
31 1
log ( ) ( ) 6 .0546468656033535950...
log log1
13
0
1
xx dx =
2 2
111
2 1
2
( )
( )!!
k
k kk
!! 1 .05486710061482057896... J166
log ( )2
2
7
24
1
2 8
1
1
k
k k .05490692361330598804...
( ) ( )
( )5 6 1
61
k
kk
HW Thm. 290 1 .05491125747421709121...
51
5
1
5 1 52 20
2
1
Lik
Hk
k
k
kk
( )
( )
1 .05501887719852386306...
284
38
4 10
log( )(
3)
k
k kkk
.0550980286498659683...
log
log
2
4
1
4 12 20
1
2
x
dx
x GR 4.282.7 .05515890003816289835...
( )
!( )!
k
k kk I
k kk k
1
1
21
1
221
2
.0551766951906664843...
( )
( )!/2
11 1
1
2 1
1
2k
kk e
k
k
k
1 .05522743812929880804...
5
64 2
1 2
4
3
21
( ) ( )!
( !)
k
kk
k k
k .05524271728019902534...
2
31
6 3
48 2 2
( )
( )
k
kk
.05535964546107901888...
82 9 .0553851381374166266...
2
22
7
6
2
21 2 2 12
2
1
1
coth ( ) ( )
k
kk
k k
k
1 .05538953448087917898...
3
32 2 1
3 2
40
2
2 20
log logx
xdx
x
x xdx 2 .05544517187371713576...
x
xx dx
2
40
1
21
1
log GR 4.61.7 =
211
4 2
1
4 1
21 2
1
( )
( )
k
kk
k
k
k
k .0554563517369943079...
1
18 .05555555555555555555 =
1
22 2 2 2 2cos cosh sin sinh 2 1 .05563493972360084358...
4
4 2 10
k
k k k( )!( )
=
1
18
2
93
1
920
csch( )k
k k .05566823333919918611...
1
4
5
4
1
4 511
log( )
H nk
kkn
...
Hkk
k 2 11
2 .05583811130112521475...
1
42k
k k( )
0558656982586571151... .
38 .05594559842663329504... 14e
0
3/13/1
9)!2(
1
2
1
3
1cosh
kkk
ee 1 .05607186782993928953...
5
24
1
1
2 2
0
log( )
( )
x
x xdx 2 .05616758356028304559...
.05633047284042820302... k
k kx d
k
1
152 4
5
log( ) x
1 8 2 8 2 5 2 5 41
52 2
2 log log log log Li .05639880925494618472...
( )
( )
1
4 1
1
21
k
kk k
=
.05650534508283039223...
e
x x
xdx4 2
01 4
tan
GR 3.749.1
3
2
3
2
1
2
3
2
1
3
1
32
21
log log( )
LiH k
kk
kk
1 .05661186908589277541...
12 2 12
22
2
22
0
1
i i i i
Li e x x dxi log ( ) log( sin ) ...
3
4
4 73 .05681827550182792733...
222
22
)1(
1
4
3csch
4coth
4 k k
...
3
42
1
1 3
2 3
21 2
log( )
( )( )
( )
( )!
k !
k kk k
k
k .05685281944005469058...
1
2 2 1 2 21 k k kk ( )( )
= J249
= 1)2(1
1
kk
k
k
.05696447062846142723... 38 1
2
1 2
1
e
k
k
k
k
( )
( )!
.05699273625446670926... 1
8 8 2 2
1
142
csch sech
( )k
k k
.05712283631136784893... 1
22 3
14 3
2
( ) ( )k kk
= ( ) ( ) ( , )
1 3 11
1 1
k
k k
k d 2z k
.05719697698347550546... 7 675
120
4 3
4 31
log x
x x
.05724784963607618844... G x dx
e ex x16 4 40
1
21 1
1
2 1 2 10
Ei Eik kk
( ) ( )( )!(
1 .05725087537572851457... )
GR 8.432.1
= SinhIntegralx
xdx
x
dx
x( )
sinhsinh1
1
0
1
1
7 3
2
2
31
( ) ( )
r n
nk
5 .0572927945177198187...
( )log
( )3
42 2
7
720 124
14 2
5 42
k
k k k .0573369014109454687...
.05754027657865082526...
e
x
xdx4 21
4
1
cos
GR 3.749.2
14
2431
21
5
1
( )kk
k
k .0576131687242798354...
1
2 33
33 3
31
Li e Li ek
i ik
k
/ / cos
1 .05764667095646375314...
log log ( )4 2
0
12
4
1
1
x
xdx .05770877464577086297...
52 5 1 52
00 0 1
k
k kI F
( !)( ) ; ;
17 .05777785336906047201...
7 6
4
3
2
2
51
( ) ( )
H
kk
k
Berndt 9.9.5 1 .0578799592559688775...
log ( )
( )
2
2 2
2 1
2 2 1
1
2
1
22 11 1
k
k kkk k
arctanh .05796575782920622441... k
)
( ) (
1 11
1
k
k
k =
log log
( )
x dx
x1 21
= GR 4.325.3
( )( )
( )k
k
k
kLi
k kk k k5
25
25
1
5 11 1
1 .05800856534682915479...
.05807820472492238243... 3
2
7
6
2 1
2 32
( )
( )!
k
k kk
!!
...
11
21 2 2 1
1
21
1
2
arccoth i
ii
iarctan arctan
= sin
sin
/ 3
20
4
1
x
xdx
3 3
2 3
( )
.05815226940437519841...
1
1
( )
!
k
kk
2 .0581944359406266129...
.058365351918698456359...
12
21
2 )1(2
)1(
3
2
2
3log4
2
12
kk
k
k
kLi
2 10
11
4
5
4
1
4
1
4 4 1F
kkk
, , ,( )
1 .05843511582378211213...
.05856382356485817586...
( )2 1
2 11
kk
k
.05861382625420994135...
e
x x
xdx4 2
01 4
cot
GR 3.749.2
4
24 , volume of unit 4-sphere 4 .05871212641676821819...
.0588235294117647 = 1
171 2
44 1
0 0
( )cos( )k k
kx
x
edx
( )
( )
( ) log4
4 41
k k
kk
.05882694094725862925...
1
18
1
9
2
3
1
9 2
1
21
21 2 1
1
log arctan ( )k kk
k
kH .05886052973059857964...
1
2
9
8
1
17
1
17 2 12 10
log( )
arctanhk
k k .05889151782819172727...
H k
kk
k
sin
1
1 .058953464852310349...
log( )
log3
4 4 4
1
41
1
42 1
k
k kkk k
.05897703250591215633... k
3 .05898844426198225439...
3 5 46
3 6 21
2 1
1 1 122
12 3 4
( ) ( ) log
( )
H k
k k k kk
k
7
11520
1
2
4 1
41
4
( )
( )Re ( )
k
k kLi i .05918955184357786985...
8
1
3
1
2 1 2 1 2 31
( )( )(k k k .05936574836539082147...
)k
J240, J627
ee
erfkk
22
1
2
1
2
!!
1 .05940740534357614454...
10 !
!!
!!
1
2
1
2 kk k
k
kerf
ee
3 .05940740534357614454...
ee
erfk
kk
12
1
2 0
!!
4 .05940740534357614454...
.059568942236683030025...
1 12
11
1
2
( )( )k
kk
k
43
720
1
1 42
k k kk ( )( .0597222222222222222222 =
)
2
0
2
161 2 2 2( log log ) log
/
x x dx .05973244312050459885...
.05977030443557374251... 10 27 4
88 2
3 3
2
1
1
1
2 41
( )
( )log
( ) ( )
( )
k
k k k
1 .05997431887... j4 J311
= 50
3
... e erfk
k kk
12
2 11
( )!!
( )!! !
=e dx
x
x
10
1
... 1 12
1
20 1
4
21
e erfe k
k
k
k
, ,!
( )!
.06017283993600197841... 5
18
1
2 2
1
2
1
2 2 1
1 2
0
arctan( )
( )
k
kk
k
k
.06017573696145124717... 2123
35280
1
3 71
k k kk ( )( )
... 5 1 6 1
161
2
4
0
sin cos( )
( )
!
k
k
k
k
... 2 62
33
3
21
2 12 1
1
2
log log ( )( )(
k kk )k
H
k k
1 .06042024132807165438... ( )ck
kkc
1
12
.06053729682182486115... 112 9 24 2
1081
22 2
0
1
loglog( ) logx x x dx
.06055913414121058628... 3
512
5 .06058989694951351915... k k
kk
0
1 2 1
( )
... 3
2 21
2 1
2 91
( )
( )!!
k
k kk
!! CFG B4, J166
= 11
2 50
( )k
k k
.060930216216328763956... 23
2
3
4
1
3 11
log( )(
2)
kk k k k
... ( )( )
arctan
12 1 1
2 1
1 11
1 2
k
k k
k
k k
k
... sin( )
( )!2
1
2 11
1
4
1
2 2
k
kk k
.06147271494065079891... 4 2
3 6
7 3
4 6
7 3
41
logcot cot
i i
=( )
14
2
k
k k k
... 1
3
9 6 2 24 2 40
36 24 2 1
2
20
4
sin
( sin )
/ x
xdx
.06158863958792450009... 1
4
304
105
1
2 90
( )k
k k
... k
kk ( )2 12 21
.06160974642842427642...
3
4 4 22 23 4
1 4 1 4 /
/ /csc csch
= ( )
1
242
k
k k
.06163735046020626998... 9 81
24
1
2
1
2 4 1
1
1
cos sin( )
( )! ( )
k
kk k k
=5
811
51
4
1
( )kk
k
k
... ( )
( !)[{},{ , , }, ]
11 1 1 1
40
k
k kHypPFQ
... 35 40 2 6 31
1
1
4 41
log ( )( )
( )
k
k k k
.06196773353931867016... 2
25
3
51
2
61
1
2
( )k
kk
k
k
k
... arccsch4
.06210770748126123903... 216 1
3 2
4 21
2 2
20
1
log logx
x xdx
x x
xdx
8 .0622577482985496524... 65
... 1
16 4 22 2
1
8
1
21
csch
( )k
k k
... ( ) ( )k kk
1 22
1
= 1
16
1
11 2 1 15 2 2
2 1
k k
k kk k( )
( ) ( )
=x dx
x( )2 30 2
= log 11
4 91
x
dx
x
... 1
1 k kk
k
... 1
221
2k
k
.062515258789062500054... 1
241
k
k
1 .06269354038321393057... 2
2log
12
22
... ( ) ( ) ( )611
90
28
384 14 3 4 5
4 2
=1
16 41 k kk ( )
.062783030996353104594... 1
0
12 )1log( dxx
... 3
22
3 2
4
2 2
2 2 22 1
1
log logkH k
kk
... 5635
6 185 3 5
1
1
2 4
5 41
( ) ( )( )k kk
.06310625275909953582... log
log( ) ( )2 1
1
2
22
1 1
1
k
k
k
k
.063277280080096903577... 32 4
19
72 0
1
G
x x dx arccot3 log x
... 4 2
4 4145
10
335
1
1
k kk ( )
... 8 41
28
1
2
1
2 4 10
sinh cosh( )! ( )k kk
k
...
4
34
01536
1
1536
1
2
1
4 2
( )
( )kk
.0634363430819095293... 11
22
3
1
2 31 0
4
ek
k k k k kk k( )! !( )( )( )
... 8
3 32 3
12
2 1 20
Gk
kk
k
log
( )
Berndt Ch. 9
... 1
2 31
k kk
( )
Ik k
k0 2
0
1
2
1
16
( !)
...
= ek
kk
( )
( !)
!
1
22
0
arctan( )
( )
1
2
2
5
1
4 2 1
1
0
k
kk
k
k ...
... 1
5
... 1
4
4
4 141
4( )
log ( )
( )
logk
k
p
pk p prime
.063827327695777400598... 2 212
1
2
12
3 22
log( )
k
k k k
... 21
2
4
62
( )
( )
... ( )1 2
3
...
3
1)1(
... 12
1 k kk
... 4
3164
7 3
42
2 1
( )log
( )
H
kk
k
... ( ) ( )
1
2
1
1
k
kk
k
... ( ) ( )
( )
3 5
42
... 5
... 9
... 1
9 3
1 2
2
3
21
( ) ( )!
( !)
k
kk
k k
k
... 2
125
28 2
125 51
2
3
1
arccsch( )k
k
kk
k
... 205
34 2 3
1
1
2
3 41
( ) ( )( )k kk
... log ( )2
3
1
6
1
3 90
k
k k
= log 11
6 71
x
dx
x
... 1
486
2
3
1
3 23
40
( )
( )
kk
...
2 2
)()1(
kk
k k
.06451797439929041936... 1
1
1
11
2 12 2
12
1e
kkk
k
( )
( )
.06471696327987555495... 4
2 2 1 32
0
1
eEi
x x
edxx ( )
log
... 1
64
1
8
3
8
5
8
7
81 1 1 1 ( ) ( ) ( ) ( )
=1
32
1
8
3
8 81 1
2
( ) ( )
= ( )
( )
( )
( )
1
4 3
1
4 1
1
2
1
21
k k
k k k
.06487901723815465273...
1
02
3
)1(
)1log(
2
2log3
332
1dx
x
x
... ( ) ( )4 6
90 945
14 6 2
61
k
kk
... Ei
e
k
kk
k
( )( )
( )
( )
1 11 1
1
11 !
... log
( )2
41
4k
kk
... 11
12
3
18
3
2
1
9 31
1
322 2
log( )
( )
k k
k
k
kk
k
... log
loglog ( )2
2 3 3
2
4
3
5
2
2
2
3
1
2
2
3
1
2
1
3
3
16
Li Li Li
=log ( )
( )
2
0
1 1
4
x
x xdx
... 52 1
2
2 21
2
k
k kk ( ) ( )
1 ... 63
2
2 1
2
3 22
( )!!
( )!! (
k
k k kk )
... G
k
x
xdx
k
k2 8
1
2 1 121
2 21
( )
( )
log
( )
.06529212022186135839... 1
2
1
21
212
1
log sec ( )sin ( / )
k
k
k
k J523
.06530659712633423604... 10
61
2 21e
k
k
k
kk
( )
( )!
... ( ) log
1 2
2
k
k
k
k
.06544513657702433167... 241
144
8 2
3
2
2
1
5
2
1
log log ( )kk
k
H
k
...
02
)1()1(
)(
)1(
2
1
24
1
k
k
k
... ( )( )
13 1
3
1
3 11
13
2
kk
k k
k
k
.06557201756856993666... 819 3
2
81 3
23 3
4
32 1
log( ) ( )
= 1
3 13 21 k kk ( )
.06567965740448090483... 2 2
3
19
36
1
2 3
1
1
log ( )
( )(
)
k
k k k k
.06573748689154031248... 7 3
128
( )
.065774170245478381098... log( )( )csc ( )
1 1 1 22 2
1
k
kkk
1
.06579736267392905746... 2
21150
2
25
1
5
( )
( )kk
.0658223444747044061... 2
136
5
24 1 2 3
H
k k k kk
k ( )( )( )
... e
k
k
k
2
016
19
48
2
4
( )!
... log2
4 8
1
2 6 5 41
dx
x x x x3
... e e
.06604212644480172199... 31
31
2 1
2 3 2 11
arcsin( )!!
( )!! ( )
k
k kkk
... 2
2 2120 5 4 5 5
1
2
1
5 1csc cot
( )
kk
J840
... 2 3 4 4 2 101
12 41
( ) ( ) ( )( )
k kk
... csc( )cos ( ) sin
1
2 21 1 3 1
1
122 2
2 21
kk
=
12
2 1)2(
2
1cot
)1)(1(2
3
kk
k
.06633268955336798335... 61
256
3
4
1
43 3
3
4 21
( ) ( ) log x
x x
... log9
10
.06640625023283064365... 1
22 p
p prime
.06642150902189314371... 1
4
2
16 1
4 2
4 121 1
4 21
k
kk
kk
k
k k
( ) ( )
.06649658092772603273... 2
3
3
4
1 2 1
2 22
( ) ( )!
( )!
k
kk
k
k
!
... 1
1 eek
k
.06662631248095397825... 1 22
2
1
1
2 2
20
1
loglog log ( )
( )
x
xdx
.066666666666666666666 = 1
15
= k k
k kk
( )
( )( )
6
2 41
J1061
1 .066666666666666666666 = 16
153
1
2
1
4 3
21
1
202
2
,( )k
k kk
k
kk
.06676569631226131157... 1
2
8
7
1
15log arctanh
=1
15 2 12 10
kk k
( ) K148
.06678093906442190474... ( ) log3
18 1
2
41
2
30
x
x xdx
x dx
e x
... k k
kk
1
1 3
( )
... 1
21 k k
k
... 2 2 3 4 3 4 2 72 1
16 11
log log( )
k
k k kkk
... 2 2
3
2 2
34
1
23 3
2 3
3
log log( )
Li
=log
( )( )
2
120
1
1
x
x xdx
... 3
15
.067091633096215553429... csc cot
( )
( )2 2
2 2 21
1
4
1
4 1
2 1
k kk
k
... 12
ii eEieEii
=
1 !
sin,0,0
21
k
ii
kk
kee
i
... 48
2 2 113
20
12G
xx dx
log log log
=169
2700
1
3 61
k k kk ( )( )
... 1
2 10 5
1
25 1
2
2521 1
cot
( )
k
k
kk
k
... 90
13311
101
2
1
( )kk
k
k
... cosh ( )( )
31
2
9
23 3
0
!
e ek
k
k
AS 4.5.63
.06770937508392288924... ( ) ( )
( )
1 2 1
16
16
16 1
1
12 2
1
k
kk k
k k k
k
...
( ) ( )( )
( )2
11 1
1
1
2
1 1
k
kk
k
k kk k
...
e
e
x
xdx
2 2
2
12 20
1
cos( log )
... 2 1 2
0e
t
tdt
cos
AS 4.3.146
...
1
62 1
1
1 22
( )
( )
( )(
k
k kk )
Adamchick-Srivastava 2.22
.06804138174397716939... 1
6 6
1
8
21 2
0
( )k
kk
k k
k
... 1
4
1
1
2
2 21
log
e
e
dx
e ex x2
... log( )
( )( )2
5
8
1
1 3
1
14 3
2
k
k
k
H
k k
dx
x x
=dx
x x( )
1 31
1 .06821393681276108077... ek e
eke
e
1
0 !
... 122
1)(
kk
Hk
...
2
4 21 26 2
11 2
coth ( ) ( )k k
kk
k
k
1
... 4
4 12 2 20
1
dx
x x( log )( ) GR 4.282.3
... 7 3 2
72 41
2
log log x
xdx
... 919
13440
1
2 81
k k kk ( )( )
... ( )k
kk 22
... cosh( )!
/ /1
2
1
2
1 1
20e
e e
k e
e e
kk
...
2
31
2
31
9
)3(833
iLi
iLi
.06853308726322481314... log loglog( )3
22
1
2
1
4
1
52 20
1
Li Li
x
x
.0685333821022915429...
sin
3
2 31
32
2 kk
2 .0687365065558226299... 2
2 1 1 1 1 1 1 2 2 2 2 2 2 25
1
k
k k kHypPFQ
!{ , , , , , },{ , , , , , },
.06891126589612537985... log
( )k
kk4
1
4
1 .068959332115595113425...
05 15
4csc
555
2
5
2
x
dx
... 8
7291
1
3 1
1
3 1
4
4 41
( ) ( )k kk
... 22
71
2 1
2 81
( )
( )!!
k
k kk
!!
... 3
29
.0691955458920286028... 2 8
2
4 0
log log sinx x
edxx
.0693063727312726561... 12 2 3 14
542
0
1logarcsin log
x x x dx
... 1
512
7
8
3
8 12 2
2
41
( ) ( ) log
x
x
... ( )3 1
3
1
3 113
2
k
kkk k
... 1
1 1 1 1 2 2 2 2 13
1 k kHypPFQ
k !{ , , , },{ , , , ,},
.06942004590872447261...
32 2 2
2
2 30
x dx
x( )
.0694398829909612926...
3 9 36
184
3
1
3 1
21
2 21
log( )
( )
k kk
.06955957164158097295... log2
1
2
1
x
xdx
.06973319205204841124... 2 3
27
1
3 12 21
dx
x x( )
1 .06973319205204841124... 2 3
27
2
3 21
22 2
3
2
1
402 1
kk
k
Fk
, , ,
3 .06998012383946546544... 3
.07009307638669401196... 5
12
2
2
1
2 81
1
201
2
log ( )( )
( )k
k
kk
kk
k
= log( )
11
1 31
x
dx
x
.07015057996275895686... 7
9720
4 3
41
log x
x xdx
... ( )
( )!cosh
k
k kk k2
1 1
21
2 1
k
... e
e x dx
1
2 0
sin x
.0704887501485211468...
( )
log logcosh3 1
3
1
33
3
21
k
kk
1 .07061055633093042768... 41
4
1
4 13
42 20
2
1
Lik
Hk
k
k
kk
( )
( )
... 1e
e
e
e
.0706857962810410866... 4 2 3 41
5 42
( ) ( ) ( )k kk
= 1
1 41 k kk ( )
179 .0707322923047554885... 2 2
2 2 5
0
23 2
18 2 1
loglog log
( )(
x dx
x x ) GR 4.264.3
.07073553026306459368... 2
9
.0707963267948966192...
3
2 21 1
33 3
1
ie e
k
ki i
k
log logsin
1 .0707963267948966192...
1
2 1
sin k
kk
GR 1.441.1, GR 1.448.1
= arctansin
cos
sin1
1 1
2
21
k
kk
2 .0707963267948966192...
1
2
22 20
k
k k
k
... 1
1000
4
5
3
102 2
2
4 11
( ) ( ) log
x
x x
... 3
8
3
4
2
3
1
2 122
log( )
( )
k
kk k
... arctan log log sin2
5
5
20
2
5
2
0
x x
edxx
... ( ) ( )
1 212
kk
k
F k 1
3 .0710678118654752440... 5 2 48
2 12
0
k k
kkk
7 .0710678118654752440... 50 5 2
... 120326 1
60
e k k
k
k
( )
!( )
.071308742298512508874... ( )
log log log ( ) log3
48 2 2 2
66 12
22
0
1
x x dx
... iI e I e
k
ki i
k22 20 0
1
sin
( !)2
... ( )
14
1 2
1
k kk
k
kF
... 16
1
8
1
4 3 4 1 4 11
( )( )(k k k )k
J239
= log
log
x
x
dx
x 2 20
1
24 1 GR 4.282.8
.0714285714285714825 = 1
14
.0717737886118051393... 3 2
3 30432
x
e edxx x
... 1
3
1
2
1
21
1
31
1
332 k kk
.07179676972449082589... 7 4 31
16 1
2
1
kk k
k
k( )
.07189823963991674726... 1
4096
3
8
7
8 13 3
3
41
( ) ( ) log
x
x
... 10 3
82 2
0
4
sin tan/
x x dx
...
1 4
1
4
1
kkLi
=9
251
91
2
1
( )kk
k
k
... ( )kk
16
2
... 8
.07246703342411321824... 2
20
4 22
9
12
1
3
1
( )
( )
( )k
k
k
kk k k
= ( )cos
1 12
21
k
k
k
k
=log
( )
x
x xdx3
1 1
=x
e edxx x3 2
0
GR 3.411.11
.07246703342411321824... 2 2
0
13
12
1 1
( ) log( )x x
xdx
.072700105960963691568... 3
8 6 3
2 1
91
( )kk
k
... 1 21
2
1
4 2 1
1
1
arctan( )
( )
k
kk k
.07277708725845669284... 1
1000
3
10
4
53 3
3
4 11
( ) ( ) log
x
x x
... lim loglog
n k
n
nk
k
1
22
1
Berndt 8.17.2
...
11
1 1
1
12 2
2( ) k kk
=1 1
21 1
2
( )kk
k
.07296266134931404078... 12
5
4
13
4
2 1
4 4 1
1
1
k
k k
k
kk
( )
( )
...
0
2
64
12
4
12
kkk
kK
... cos2
10
x
edxx
.07325515198082261749... 1
20736
1
4
3
43 3
3
4 21
( ) ( ) log
x
x x
...
1
1
4 !
4)1(4
k
kk
k
k
e
... ( )( )
, , ,
11
2
1
81 2 3
1
222
21 1
kk
k k
k
k kF
k
=
16
9log1
2
1
... 120 2 2670 2 8520 2 8000 26 7 8 9I I I I( ) ( ) ( ) ( ) 3025 2 511 2 38 2 210 11 12 13I I I I( ) ( ) ( ) ( )
=k
k kk
4
21 5( !) ( )
... 7
4
...
2
2cosh
... 2
2 21
2 3
6
1
12 7
1
12 5
( )
( ) ( )
k kk
J346
1 .07351706431139209227...
1
2
)!2(
)2()!(
k k
kk
...
3 3
2
3
3
10
1
3 80
log ( )k
k k
...
12
1
)12(
)1(
k
kk
k
H
.07363107781851077903... 3
128
1
4 2 1 2 3 2 5
2
0
kk k k k
k
k( )( )( )
... ci( )
... 5
2
4 1
2 2
1
1
e k
k
kk
( )
( )!
.07388002973920462501... 35
48 2 CFG B5
.07394180231599241053... 8
3
11
3 24
0
4
sin tan/
x x d
x
.07399980675447242740...
2
2
2
21
2
1sinh
k
kk
1 .07399980675447242740... 12
21 1
sinh( ) ( )( ) ( )i i
... 3 62
33
2
3 3 12
1
log log( )(
H
k kk
kk 2)
.07402868138609661224... log log log
(log )( )2 3
2 3
2 3
2
2 2
32
1
2
1
2
5 3
8
Li Li
= log ( )
( )(
2
0
1 1
1 2
x
x x x )
.074074074074074074074 = 2
271
21
22 2
2
0
3
02 3
( ) ( ) ( ) ( )kk
k
kk
k
k kLi Li
= log log2
41
3
41
x
x
x
x
1 .0741508456720383647...
2
51
5
10
2( )
( )
( )
k
k k Titchmarsh 1.2.8
.07418593219600418181... ( )( )
11
1 13
23
2
k
k kkk Li
k k
2 .07422504479637891391...
155/45/35/25/1
11
)1()1()1()1(
1
k k
... 1
3
5
4 3 1 31
log( )(
dx
x x 2)
... 1
2
14
7
7
2
1
2 720
csch ( )k
k k
... 21
2
1
2 1 23 30
3
1
Lik
Hk
k
k
kk
( )
( )
=log2
21
2
02 2
x
x xdx
x
edxx
1
1 .07446819773248816683...
e ek e k
k
2 22
0
11
1log
( )
.07454093588033922743...
1
22
1
1
)1(
2
1csch
2
1
k
k
k
... Li2
4
5
...
2
12
04
0
1
16 2
1
16
1
4
1
4 1 1
G
k
x dx
xk
( )
( )
log
... ( ) ( ( ) )( )
1 4 11
11
14 4
2
k2
k k
k kk k
...
2
33
2
33
3
1
6
1
3
2 ii
= 2
3
1
6
1
31 1
11 3 2 34
2
( ) ( )/ /
k kk
= ( ) ( )
1 3 11
1
k
k
k 1
.074877594399054501993... 35 3
72
1
41
2
21
3
0
1
k kx x
k ( )log( ) log x dx
.07500000000000000000 = 3
40
1 .07504760349992023872... e
=5
66 5 B
2 .07510976937815191446... 1
40 ( !!)kk
.07512855644747464284... ( ) log( ) log3
16
1 4
0
1
x x
xdx
1 .07521916938666853671... 22
5
1
5( ) ( )
d k kk
Titchmarsh 1.2.2
.07522047803571148309... 1
4
1
2
1
2
1
2
1
2 1 2
1 2
1
cos cos( )
( )!
k
kk
k
k
1 .07548231525329211758... 5 3 3 2 ( ) ( )
... ( ) ( )( )
( ) ( ) ( )( )
2 34
42 3 2 5
1 41
H
k kk
k
... 14 3
22
1
27 3
1 3
31
log ( )k
k
k
k k Berndt 2.5.4
... 1
8
1
2
1 3
1e
k
k
k
kk
( )
!
.07591836734693877551... 93
1225
1
2 71
k k kk ( )( )
... 2031
2
1
4 21 4
1
1
ek k
k
kk
/ ( )
! ( )
.07606159707840983298... 190 4 1
44
( )
( )
... e
erfik
k
k k
k
1 11 4
2
1
1
( ) !
( )!
... e
erfi
kk
k
dx
e x
k k
kx1
1 42 1
1
1 0
1
( )
!
... ( )
log2 1 1
2 1
1
2
1
1
1
1 2
k
k
k
k kk k
= 12
2
1
2
log
arccoth kkk
.07646332214806367238... 1
41
21
2
3
2
3
2
i i i i
1 21
32
log( )k
k k k
... log ( )( )
237
607
1
70
bk
k
k
GR 8.375
= dx
x x8 71
... 12
1
2
1coth
2 2
e
J983
= 1
1
1
21 2 1 12
1
1
1kk i
k
k
k
( ) ( ) Im ( )
= sin x dx
ex
10
= dxx
x
1
0
1
1
logsin
...
2
1
2
1
2 12coth
e
= 1
1
3
21 2 12
0
1
1kk i
k
k
k
( ) ( ) Im ( )
.07668525568456209094... 1 5
25 2/
.07671320486001367265... 1 2
4 1
3
20
2
log sin log(sin )
sin
/ x x
xdx
GR 4.386.2
.07681755829903978052... 56
7291
81
2
1
( )kk
k
k
=1
13
... G G 2
... pk
kk
p prime k
4
1
4
( )
log ( )
... ( ) ( , )( )
( , , )39
83 3
1
31 3 0
30
kk
=1
2 1
2 2
0
1 x x
xdx
log
GR 4.261.12
=
1
0
1
0
1
0
222
1dzdydx
xyz
zyx
.07707533991362915215... 1
2
7
6
1
13
1
13 2 12 10
log( )
arctanhk
k k
... 2 2
40128
2 1
4 2
( )
( )
k
kk
=139
1800
1
3 51
k k kk ( )( )
.07720043958926015531... log3
41 1
x
xdx
.07721566490153286061...
1
2
1
120
1
log ( )x
x
xdx2 GR 4.283.4
= 21 12 2
0
x dx
x e x( )(
)
x
Andrews p. 87
... 1 10
1
( )x d
.07730880240793844225... 4
12604 T( )
... 9
162
3
2 12 1 1
2
2
log( )
( )k
kk
k
Adamchik-Srivastava 2.24
... 11
2
1
4 2 1
1
1
erfk k
k
kk
( )
! ( )
1 .07746324137541851817... 1
31 k kk !!
.07752071017393104727... log( cos ) ( )cos
2 1 121
1
k
k
k
k GR 1.441.4
.07755858036092111976... ( )k
k kkk k5
1
25 522
1
5 .07770625192980658253... 9 1 2 /
... ( )
( )arctan log
1
2 1 44
1
2
5
42
1
1
k
kk k k
3 .07775629309491924963... 0
2
1
0
112 2
( ) ( )k mk
n
kmn
mn
.07777777777777777777 = 90
7
.07784610394702787502...
cos( )
!e
e
k
k k
k
1 1
0
.07792440345582405854... 2 2 1 1 11 2 11
cos sin ( )( )( )
k
!k
k
k k
... 10514
4
p
p prime
Berndt 5.28
=
( )
( )
( )4
8 41
k
kk
Titchmarsh 1.2.7
= qq squarefree
4
.07795867267256012493... 8
92
2
3
1
2 1
1 2
0
log( )
( )
k
kk
k
k
... 50 43
128
1
7
1
2 7
2 1
21
2G
k
k
kk
( )!!
( )!! J385
... 1
log
... ( )
!/k
ke
kk
k
k
2
1
1
11
...
2
3
13 20e
x
xdx
cos
... 1
12 2 2
12 22 2
1
2 2 21
csch sech
sinh( )
( )
k
k k k
...
2
1
2
1
21
21
iiii
...
4 3
3
8
1
3 2 1
1
1
( )
( )
k
kk
k
k
...
sin sinh
cosh cos
2 2
2 2 2 21
= 1
11 44
2
1
1kk
k
k
k
( ) ( ) 1
=17819
226800
1
1 101
k k kk ( )( )
.078679152573139970497... 2
32
26 1
2 1
21
log ( )( )
k
kk
1
=
kk
kkk
32
2
11 1
2log
... ci x x dx( ) log cos log log sin1 2 1 2 20
1
... 121
kk
k
...
1 2
1)12()1(6
22log
3
2
k k
k
... 1
124 24 2 27
1
2 22 2
21
log( )k
k k
... 3 3
4 12
1
1
2 1
31
( ) ( )
( )
k
k
k
k
...
primep p 41
1log)4(log HW Sec. 17.7
... log ( )( )
4
3 31
32
kk
k
k
k Dingle 3.37
.07921666666666666666 = 7
961
12
19
logx
dx
x
... 2
)3(3
24
)6(53)2,1,3(
2MHS
... 1
2
1
22
1
2
1
2 1 21
cos sin( )
( )!
k
kk
k
k
... 3 2 21
1 21
log( )
( )(
)
k
k
k
k k
=1
16
2 1
1631 1k k
k
kk
k
( )
.07957747154594766788... 1
4
...
1
125
3
52 ( )
.07998284307169517701... 32 3
3 3
2
1
3
4
3
1
3 11
21
log
( )( )
k kk
= log( ) log1 3
0
1
x x dx
.080000000000000000000 = 2
25 10 11
( )kk
k
.080039732245114496725... 2 3251
10842
2
5 41
( ) ( )log( )
x
x x
= x x
xdx
3
0
1
1
log
2 .08008382305190411453... 3 92 3 3/
... 7
61
2 1
2 71
( )
( )!!
k
k kk
!!
.08017209138601023641... 4 3
411215
log x
x xdx
.0803395836283518706... log
( )(
3
2 6 3
1
6
1
3 2 3 31
k kk )
= ( )
( )
k
k k
dx
x x xkk k
1
3
1
3 3 12 26 5 4
1
.08035760321666974058... log log2 2 2
.08037423860937130786... 11 5 14 1
2
1
1k kk
k
k
k
( ) ( ) 1
... 1
3696 2 18 2 55 1
42 1
1
log log ( )
k k
k
H
k
... 1
1 113 2
( ) ( )k mk
n
kmn
mn
... log
( )
3
2 6 1
k
k kk
... 2 11
2cos e ei i
.08065155633076705272... 1
4
1
2 1 222
1
( ) ( )e
k B
kk
k !
.080836672802165433362... 10
50 10 5 51 2
25
0
1
x
xx dxarctan( )
... ( )
log
14
2
k
k k k
... 1 21
1
1
21
2
log ( )k k
kk
K ex. 125
= GR 6.441 log ( ) x dx1
2
=
1 1
1
1
20
1
log logx x
dx
x GR 4.283.2
122 .081167438133896765742... 8
63120 6 1
1
65
5
0
1
( )log( ) x
xdx
.08126850326921835705... 13
15 4
1
2 708 6
1
6
0
4
( )
tan/k
k k
dx
x xx dx
.08134370606024783679... k
kk
k
k
k
2
62
1
111 6 2
( ) ( ) 1
.08135442427619763926... 9 21
31
2
2 1 9 2 11
arcsin( )!!
( )!! ( )
k
k kkk
2 .08136898100560779787... 1
2 220log
x dx
x
.08150589160708211343... 1
22
3
4 11
loglog
( )(
dx
e e ex x x 2)
1 .08163898093058134474... 3
3 30
3 3
321
1
2 1
1
2 2
( )( )
( ) ( )k
k k k
.08183283282780650656... k
kk
k
k
k
3
72
1
111 7 3
( ) ( ) 1
1 .08194757986905618266... 4 1
2
1
16120
sin( )
!( )
k
kk k k
.08197662162246572967... 153
26
3 11
log( )(
2)
k
k kkk
.081976706869326424385...
1 )!2()1(2
3
2
1
1
1
k
k
k
B
e
e
e
1 .081976706869326424385... 1
2
1
2 2
1 2 1 2
1 2 1 2coth/ /
/ /
e e
e e
= B
kk
k
2
0 2( )!
AS 4.5.67
.082031250000000000000 = 21
2561
71
2
1
( )kk
k
k
.08217557586763947333... 1
2 6 3
2
6
1
6 809 3
1
log ( )k
k k
dx
x x
.08220097694658271483... log
log loglog2 2
2 2
2
22 3
3
2
1
3
1
2
Li Li
.0822405264650125059... 2 2
2212
2
2
1
2
1
2
logk
k k
1 .08232323371113819152... )1,1,2()4(90
4
MHS
1 .082392200292393968799... 4 2 2 11
4 60
( )k
k k
.0824593807002189801... 7
.08248290463863016366... 52
34
1
8
2 1 1
8 1
22
0 0
( ) ( )
( )
k
kk
k
kk
k k
k
k
k
k
k
.08257027796642312563...
1
16 8
2
8
2 2
2 2
cothsin sinh
cos cosh
= ( )8 4 11
14 4
2
kk kk k
.08264462809917355372... 10
121
1
10
1
1
( )k
kk
k
1 .08269110766346817971... H
k
ek
k !
1
6 .082762530298219680... 37
.08282126964669066634... ( )7 3 11
14 3
2
kk kk k
... ( )
log( )6 2 1
11
2 6
2
k
kk k
k k
.08285364400527561924... 328
)3(7
4
3
64
1 3)2(
= )()(28
)3(733 iLiiLi
i
= dxx
xdx
x
xx
14
21
04
22
1
log
1
log
...
4/14/14/14/14/1 2
1sinh
2
1cos
2
1sin
2
1cosh
216
1
4/14/1 2
1sinh
2
1sin
28
1
=
1
21
)!4(
2)1(
k
kk
k
k
15 .08305708802913518542... 41
11
12 1
1
7 6
1
14
1e
k k
k
k
kk
k
k
k
( )( )!
( )( )
( )! Berndt 2.9.7
122 .08307674455303501887... log
( )( )( )
5
2
5
21
k
k km
k m
.0831028751794280558... 5
2
5
6
1
5 12
1
1
log( )
( )
k
kk
k
H
k
.08320000000000000000 = 276
31251
4
4
1
( )kk
k
k
.08326043967407472341...
2 3
3
2
18
2
32
1
2
15
363
log log( )
Li
= log ( )
( )
2
0
1 1
3
x
x xdx
.083331464003213147... ( )k
kk4
2
.083333333333333333333 = 1
126 2 1
1
14 2
2
( )kk kk k
2 .083333333333333333333 = 25
124 2 1 14
1
H kk
k
( )
1 .08334160052948288729... k
kk
!!
( )!21
.08334985974162102786... k
kk
3
51 3
257
322 9 3 27 4 27 5
( )( ) ( ) ( ) ( )
1 .08334986772601479943... k
kk
!
( )!21
.08335608393974190903... log loglog( )5
32
2
3
1
3
1
42 20
1
Li Li
x
x
.08337091074632371852... ( )4 1 1
21
22
4
k
kLi
kk k
2 .08338294068338349588... cos cosh( )
1 1 21
40
!
kk
.08339856386374852153... 1 22
1
k
k
1 .08343255638471000236... 1
22 2
2
44 4
0
cos cosh( )!
k
k k
.08357849190965680633... 1
4 11k
k
( )
1 .08368031294113097595...
1
21 1 1
12
1 4 3 44
2 sin ( ) sin ( )/ /
kk
.08371538929982973031... 47 2
2
21 2
2
85
4 2 1 2
2 2
1
log log
( )( )(
3)
k H
k k kk
kk
57 .08391839763994994257... 21e 8103 .083927575384007709997... e9
1 .08395487733873059... H ( )/
33 2
.0840344058227809849... 11
2 1 21
Gk
k
k
( )
( )
= k k k
kkk k
( )
( )
2 1
16
16
16 112 2
1
= 1
8 21
3
41
kkk
k
, Adamchik (29)
= log x
x xdx4 2
1
.08403901165073792345...
2 21
21
21 2
1
log sin ( )( )!
k k
k
B
k k AS 4.3.71
.08427394416468169797... 8 32
1
4 3 4 2 4 1
2
21
( )( ) (k k k )k
J271
.08434693861468627076... ( )kk
2
2
1
.08439484441592982708... ( )5 1 11
14 1
2
kk kk k
.084403571029688132... ( )
!
4 11
4
21
k
kek
kk
... 2
1
2
1
2
1
2
1
4 21
HypPFQx
xdx, , ,
cos
.08444796251617794106... logsinh
( )log( )
4 4 11
1
4
2
k
kk
k k
.08452940066786559145... ( )3 1 1 1 1
21
2 32
k
k kLi
kk k
.084756139143774040366... 11
2
1
2
1
2
12
1
cot( )
( )!
kk
k
B
k
.0848049724711137773... e i ei i 2 24/ log log
...
4 2
24
415 42
2
26
1
2
21
43 2
log
log( ) logLi
=log ( )
( )
3
0
1 1
1
x
x xdx
.085000000000000000000 = 200
17
... 21
2 4 1
2
20
4
arctansin
cos
/
x
xdx
4 .0851130076928927352... k
dkk
2
... dx
x( )1
2
... 178
225
1
6
2 1
2
1
2 61
2
( )!!
( )!
k
k kk
J385
.085409916156959454015... 1
0
222
log)1log(2
)3(72log4
212 dxxx
.08541600000000000000 = 41
480
1
2 61
k k kk ( )( )
...
5
241
1
1
2 2
30
log( )
( )
x
x xdx
...
5
24
1
1
2 2
20
log( )
( )
x
x xdx
... 21
2
1
2 1 20
sinh( )!
k kk
= 11
2 2 21
kk
...
1 )14(4
1
kkk k
.0855209595315045441... 5
4
1
31
1
2
1
21
Eik k
k
k
( )( )
!( )
20 .08553692318766774093... e3
1 .08558178301061517156... 2
2 2 12
1
k
k kk ( )
Berndt Sec. 4.6
=
1
2
2
)(
kk
ku
... log k
k kk4 2
2
.08564884826472105334... log ( )2
3 3 3
3
4
1
3 708 5
1
k
k k
dx
x x
.08569886194979188645... log2
3
1
6
1 3
12
3 3
4
5 3
4
i i i
1 3
12
3 3
4
5 3
4
i i i
=( )
1
132
k
k k
.08580466432218341253... ( )
( )
1
22
k
kk k
1 .08586087978647216963...
3 2
2
3
4log GR 8.366.5
.08597257226217569949... 1
2
1
1 11
1
sinh
cosh cos
cos
k
ekk
.08612854633416616715...
3 1
31360
1
( )
sinh
k
k k k
1 .08616126963048755696... ee kk
k
12 2 1 2
1
2 1 10
cosh( )!( )
3 .08616126963048755696... ee kk
12 1 2
1
20
cosh( )!
.086213731989392372877... 1 1
8
1
2
1
22
12
1
sincsc
( )
( )!
kk
k
k B
k
... 2 22 2
0
2
48 8
2
4
2
4
log loglogsin sin
/
x x dx
.086266738334054414697... sech
1 .086405770512168056865... 7 3
41 2 2 2
64 4 2 1
4 2
31
( )( log ) log
( )
H
kk
k
.086413725487291025098... log cos sin
cos sin
/2
4 2 0
4
G x x
x xx dx
= GR 4.224.5 log cos/
x dx0
4
.086419753086419753 = 7
81 9 11
( )kk
k
.086643397569993163677... log 2
8
.086662976265709412933... 7
8 4
1
14 1
42 1
coth ( )k
kk k
.086668878030070078995... ( )
!
3 1 1 11
1 2
3k
k ke
k k
k
1 .086766477496790350386... 83
8 2 4 22 1
4 1
22
03
log log
( )
k
k kkk
.086805555555555555555 =25
288 1 2 3 4 5
3
1
k
k k k k kk ( )( )( )( )( )
.08683457121199725450... 1
1 2 30 k k k kk ( )!( )!( )!
.08685154725406980213... 1
8 32
2
2 1
2
31
log log
( ) ( )
x
x xdx
1
.086863488525199606126... ( ) log3 1 1 1
1
3
2
k
k
k
kk k
1 .08688690244223048609... log csc( )
1
2
3
2
3
2
3
2
2 1
1
k
k
k
k
.087011745431634613805...
642 2 1 2 2
0
2
( log ) log(sin ) sin cos/
x x x dx
... ek e
ek
kk
2
0
2/
!
... 2 1 2
4
1
4 2 1
21
1
arcsinh
( )
( )
k
kk
k
k
k
k
... 18
3 2
4
1
4
1
41
421 2
log( )
( )
k k
k
k
kk
k
=dx
x x x x5 4 3 21
... 30
3431
61
2
1
( )kk
k
k
... ( )311
96
6 8
6 85 4 31
k
k k kk
... 1
2
1
22
1
2 2 1 21
cosh sinh( )!
k
k kk
.087764352700090443963...
212 4
)4(
)14(4
1
14
1
kk
k
kkk
kk
...
12 )32(4
)2(
2
)3(3
18
1
3
log
kk kk
k
... logG
... 2 2 3 21 23
1
( ) ( )( ) (
)
k
k kk
... 2 2 33 1
2 1
1
23 21
24
( ) ( )
( )
( ) (
)
k
k k
k k
kk
k
... 2 2 71
2 2
12
2 2 11
arcsin
( )k
kk kkk
... 2465 1
50
4
21
e k k
x
xdx
k
k
e( )
!( )
log
... 6
2
3
2
31
16 4
1
log
logx
dx
x
1 .088000000000000000000 = 136
125
2 2
0
x x
edxx
sin
.08800306461253316452... J
k kk
k
k1
0
2 3
31
3
1
( )!( )!
.088011138960...
14
)(
k k
kpf
.08802954295211401722...
1
6
7
21
7
2 1 21
cos( )
kk
.08806818962515232728... 4 3
4 2116
6
log x
x xdx
6 .08806818962515232728... 4 3
40
3
20
1
16 1 1
log logx dx
x
x dx
x
=
0
3
xx ee
dxx
1 .08811621992853265180... 4
12 2 2
4 14
42 1
sinh
exp( )
k
ki i
k
kk k
.08825696421567695798... Y0 1( )
... 23
72 31
1
1
2
40
1
loglog( )
( )x
x
xdx GR 4.291.23
.088424302496173583342... where S is the set of all non-trivial integer powers
( ) 1
S
.0884395847555822353... ( ) ( )k kk
12
.08848338245436871429...
1
24
1)2,4(
jk jkMHS
... 1
25 1 3 1
1
2 2 3
1
1
cos sin( )
( )!( )
k
k
k
k k
.08856473461835375506... log tan ( )( )
( )!2
1
21
2 1
21
2 12
1
kk
k
k
B
k k AS 4.3.73
.08876648328794339088... 1
2 241
22
0
1
x x xlog( ) log dx
x
= x x x dlog log( )10
1
1 .08879304515180106525... log
( )
2
2
2 1
4 2 102
k
k kkk Berndt 9.16.3
= arcsin
logsin/x
xdx x dx
0
1
0
2
= x x
xdx
arcsin
1 20
1
=
log x
xdx
1 20
1
GR 4.241.7
= log1
1
2
20
1
x
x
dx
x GR 4.298.8
= log( GR 4.227.3 tan )/
20
2
x dx
= lo GR 4.227.15 g(tan cot )/
x x0
4
dx
= log(coth )coth
xdx
x0
GR 4.375.2
.088888888888888888888 = 45
4
... ( )
( )
k
k k k kkj j
jk
1
1
12 22
= ( ) ( )
1 121
k
mk
mk m
.08904125208589587299... 2 3
27 1
2
41
2
31
( ) log
x
x xdx
x dx
e x
= dxx
xx
1
03
22
1
log
.08904862254808623221... sin cos sin cosk k
k
k k
kkk
3 4
11
.0890738558907803451... 11 1
2
21e
erfe
xdx
x
( )
.089175637887570703178... ( )
( )
k
kk Li
kk k
1
1
112
22
2
1 .0892917406337479395...
1111 )!1(!)!2(
!)!12(2
4!1,2,
2
3(F
2
1
kkk kk
k
k
k
k
k
... ( )( )
12
1
2
1
222
21
kk
k k
k
kLi
k k
... 2
3
25
4
33
4
1 4
0
1
x dx
.08944271909999158786... 1
5 5
1
16
2
0
( )k
kk
k k
k
... dx
x k
dx
xe
k ( ) ( ) ( )0 0 0
1
1
... 2 2
7
319
2940
1
7
1
21
log ( )
k
k k k
... 1
3 16
3
8 16
7
8
2
4 8
2
4
3
8
cot cot logsin log sin
=( )
1
4 708 4
1
k
k k
dx
x x
... 10
38
2
3
1
2 20
log( )
( )
k
kk
k
k
... 90
180
1
2 441
4
( )
( )f k
kk
Titschmarsh 1.2.15
... ( )k
kLi
k kk k
1 13
23
2
1
... 4
... erfik kk
k
1
2
1
4 2 10
! ( )
=9
1001
91
( )k
kk
k
... ( )
( )log
k
k kkk k2 1
21
1
222 1
k
... 1
2
2
2
1
8
1
2
1
2
log i i i i
= sin
)
2
0 1
x
e edxx x
... ( ) log( )
2 2 21
2 22
3 31
Lik
kkk 1
... dxx
x
04
45
1
log
2512
57
... 2
11 5 5
15
... 5 5 1
2
1
5
2 1
0
kk
k
k
... 11 5 5
25
... 47
602
1
607 6
1
log( )k
k k
dx
x x
=13
144
1
3 41
k k kk ( )( )
... coth2
... 16 2 102 2
2
2
log( )
k
kkk
.09039575736110422609... 11
3
1
2
4
31
2 1
4 3 12 1
1
1
Fk
k k
k
kk
, , ,( )
( )
... 3 3
343 7
... 1
9
2
3
1
4
1
3 11
22
( )
( )
kk
= ( ) ( )k k
kk
1 1
32
... 2 41
2
4
3
1
4 2 11
arctanh log( )k
k k k
... csch( )kk
1
... 1
2 2
1
22
log
( )k
kkk
=
log 11
2
1
22 k kk
5 .090678729317165623... 1 10 1
1
22 4
1
1
2F ( , , )
( )!
k
k
k
c
kk
k
k !
... 11
111
ee
dx
e ex xlog( )( )
.090857747672948409442... e e
e
k
e
k
kk
( )
( )
( )
1
1
13
1 2
1
=1
11
=12
11 63
1
F kk
k
... 2
7
... 2
31
7 3
16 2 1
( )
( )
k
kk
... 1
2
6
5
1
11
1
11 2 12 10
log( )
arctanhk
k k K148
1 .091230258953122953094... 21
21
2
1
3 320
tan
k kk
... k kk kk k
( )( )
4 11
113 4
2
2
... 2
20
41108
1
3 3
( )
( )
logk
k k
x
x x
.09140914229522617680... 527
2 3
2
1
ek
kk
( )!
... 3 2
4 8
7
36
8
16
1
4
2
2
log ( ( ) )
G k kk
k
...
1 4
cos
1cos817
11cos4
kk
k
... 1
242 kk
... 1
G
... k k
k k k kk k
2
21
21
2
11
1
1
... 1
32 2
3 3
2
3 3
2
i i
= 13 1 14
2 1k kk
k k
( )
... 1
83 2
1
22
1
2 2 1 2
2
1
sinh cosh( )!
k
k kk
... 1
25
4
5
1
5 11
21
51
( )
( )
k
dx
xk 1
... 1
120
cos x
ex
=250
23
=3617
510 8 B
.09216017129775954758... log
log2
3
5
361
12
17
x
dx
x
.0922121261498864638... 245
64
5 4
4 21
log x
x x
... 4
2 11 2 10
log ( )( )(
k
)k
k
k k
=
122
1
02
2
)1(
11log
)1(
)1log(
x
dx
xdx
x
x
... log
( )(
3
4
3
36
1
3
1
3 1 3 31
)
k kk
= 5
541
51
2
1
( )kk
k
k
.09259647551900505426... ( )2
21
k kk
k
1 .092598427320056703295... ( )3 11
kk
...
1
1
)2)(12(
)1(
12
172log84
k
k
kkk
... ( ) log( )2 2 1 1
1
2
22
k
k
k
kk k
1 .09267147640207146772... 8
7
1
2 2 2 1 2
2
0
arcsin( !)
( )!
k
k kk
1 .09268740912914940258... 1
732 kk
... 61
32
33
2
3 21
Lix
x xdx
log
6 .093047859391469... k k
k
e k
k kk
k
k
2
2
1
21
1 1 ( )
!
( )/
.09309836571057463192... 1
2 10 20 5
10
10
cot
log
= 1
5
2
5 5
1
5
4
5
2
5cos log sin cos log sin
= ( )
1
5 708 3
1
k
k k
dx
x x
.0932390333047333804... 1 10
3
22 4
1 2 1F ( , , )
( )
!
k
k k
k
k
... ek k kk
21
8
1
2 40 !( )( )
... 18 13
32
1
5
2 1
2
1
2 51
G k
k kk
( )!!
( )! J385
.09334770577173107510... 22
8 12
13
1e
k
k kk
k
( )!( )
... pf k
kk
( )
31
.09345743518225868261... 5 6 2
9
1
2 1 2 31
log
( )(k k k )k
=3
321
33
8 11
2
12
1
( ) ( )( )k
kk
kk
kLi
k
... 1
232 5 1
1
12
log ( )( )
( )
k
k
k
k k
...
1
121
1 )!12(
)2(2)1(
2sin
1
k
kk
k k
k
kk
.094033185418749513878... 7
1920
1
4
1
16
4
42
42
44
44
Li e Li e Li e Li ei i i i( ) ( ( ) ( )
= ( )cos
1 14
41
k
k
k
k
... 9 3
4
2
2
1
6
2
4 2 321
( ) log ( )
( )
k
kkk
... 1 2 2 2 11 2
2 1
1
log log ( )( )(
k k
)k
H
k k
.09419755288906060813... 1
2
1
2
1
1
1
2 21
csc ( )k
k k
1 .09421980761323831942... 1 4
16
4 1
4 1 1
4 1
4 1
4
2
2
21
2
2
/ ( )
( )
( )
( )
k
k
k
kk
1 J1058
.09432397959183673469... 1479
15680
1
1 81
k k kk ( )( )
2 .09439510239319549231... 2
3
1
2
arccos
.09453489189183561802... 1
2
2
3
1
31
logk
kkk
J145
2 .09455148154232659148... real root of Wallis's equation, x x3 2 5 0
.09457763372636695976... ( )k k
kk
2
3 11
.094650320622476977272... Re ( )( ) ( )
ii i
2
2 .09471254726110129425... arccsch1
4
.094715265430648914224... 1
2
4 2 1
2
4 1
2 1 2 221
4
( )!!
( )! ( )( )
k
k
k
k kk
J393
.094993498849088801... 1
2 22
log k
kk
Berndt 9(27.15)
.09525289856156675428... 22
24 2 2 2 2
2 1
2 2 2 11
log log
( )!!
( )!! ( )
k
k k kkk
.095413366053212024... 8 21
2
1037
105
1
2 2 71
arctanh
kk k( )
1 .09541938988358739798...
1
2
3
4,
1 .095445115010332226914... 6
5
1
24
21
2 1
2 60 1
kk
kk
k
k
k
k
( )
( )!!
!!
... ( )
121
k
k k
k
... e Ei eEie
x
e xdxx( ) ( )
( )
2 11
1
2
0
1
... ( )1 1
3
... ( )1 4
3
.09569550305604923590...
( )3
2 11
kk
k
.09574501814355474193... 1
833 kk
.09587033987177004744... 5
3 2 41
tanh log xdx
x
... 2
2 2 4 1 2
0
1
G xlog log( ) log xdx
4 .09587469570927995876... 22
2 11
20
1
Gx
x dx
log log log
... 1
3
4
3
1
3 411
log( )
H nk
kkn
=12
1251
41
2
1
( )kk
k
k
.09601182917994165985... 5 5
54
4
41
( ) log
x
x xdx
... 1
64
3
8
7
8 11 1
41
( ) ( ) log
x
xdx
... 6
... ( )3
.09655115998944373447... 2 5 2 31 4
1
( ) ( ) ( )( )
H
kk
k
=
2
1
14
24
11)1,1,3()1,4(
k
k
jk jkjkMHSMHS
... log ( )
( )
2
2
1
4
1
4 1
1
1
k
k k k J368
= ( ) ( )
( )( )
1
2 6
1
2 2 2 40 0
k
k
k
kk k k
= 1
2 11k 2k k k k( )( )
GR 1.513.7
= dx
x xx x dx x d7 5
1
2
0
4
5
0
4
sin tan tan/ /
x
= log
( )
log
( )
x dx
x
x x d
x
1 131
30
1 x
= log 11
4 51
x
dx
x
= log
( log )(
x dx
x x 2 20
1
1 )2 GR 4.282.4
.0966149347325... ( ) log
1
11
k
k
k
k
... 2
0
2
09 2 1 2 1
2
2 2
k
k k
k
kkk k
!
( )( )!!
( !)
( )! J277
= J313 W2
= 1
6 5
1
6 11 k kk
J338
=log x dx
x60 1
... 2 3
3 3
2
3
2
32
2
32
1
2
log log
Li Li
=log
( )( )
2
12
320
1 x
x xdx
... e
e
k
ekk
2
121
( )
... ( )
( )
1
2 2 1
1
1
k
k kk
k
... e
k k
k
k
k
k
2
1 1
3
4
2
2
2
5
( )! !(k )
... e
k k
k
k
2
0
1
4
2
2
!( )
... kk
k
k
kk
k 4
1arcsin
)14(
144
22log3
2)1(
12
1
... 1
342 kk
.0974619609922506557... 1
37268
7
8
3
8 14 4
4
41
( ) ( ) log
x
x
... 2 5 2
8 2 2
1
4 2
1
21
sincsc sec
( )k
k k k
=22
225
1
2 51
k k kk ( )( )
... 9731
30
1
1 61
ek kk
( )!( )
...
iLi e Li e
k
ki i
k2
32
32
32
1
( ) ( )sin
... 312
3 21
4 2
2
3 21
log
k
k kk
= ( )( ) ( )
11
22
kk
k
k k
.0981747704246810387... 32 4
3 3
12 2
0
sin cos
( )
k k
k
dx
xk
1 .09817883607834832206... csc(log )
2ii i
.09827183642181316146...
log
5
4
.09837542259139526500... 166
3192
3
4 41
3
log( )
k
kkk
... 13 3
41 3
13
1
( )( )
( ) k
k k
=
1
0
1
0
1
0 1dzdydx
xyz
zyx
...
( )k
k kk4 2
2
... ( ) ( )2 2 1
2
2
1
k kk
k
... log( )
3 21
2
1
4 2 10
arctanhk
k k
= 11
27 331
k kk
[Ramanjuan] Berndt Ch. 2, 2.2
= Lik
k
kk
11
2
3
2
3
J 74
= sin
cos
x
xdx
20
1 .09863968993119046285... 31
3
1
3 12
32 20
2
1
Lik
Hk
k
k
kk
( )
( )
1 .098641964394156... Paris constant
.09864675579932628786... 1
2
2
42
0
1
si
x x d( )
log sin x
1 .098685805525187... Lengyel constant
.0987335167120566091... 2
2 2224
5
16
1
1
( )
( )
k
k k J401
.098765432098765432 = 8
811
81
4
41
( )logk
kk
k x
xdx
.098814506322626381787... 8 4
1
2
1
16 121
coth
kk
... ( ) ( )k kk
3 12
5 .09901951359278483003... 26
... log 2
7
... log
( )( )
( )
2
4
1
3 31
1
22
k
k kk
... ( )( ( ) )
11
2
2
2
kk
k
k
7 .0992851788909071140... k kk
kk
k
2
1
2
12 1 2
( )
... 9
... 7715
17282 5 3
5 31
( ) ( )( )
k
kk
... ( )( )
( )!
12 1
21
1
k 1
k
k
k
... 16
9
2
34 2
14
341
log
( )
( )!
k
k kk
!
... 24
51
4 12
1
log ( )( )
k kk
k
H
k
...
4
2
8
1
2 23
0
4
log
tan/Gx x dx GR 3.839.2
.0996203229953586595... 3
21
1
12
e Eik kk
( )( )!
... csc2
... 23
11
1 1
1 2
0
eEi
k
k k
k
k
( )( )
( )( )!
... e Ik
k
k
k
0
12
20
12
( )( )!
( !)
...
2
)(1log
kkk
k
.100000000000000000000 = 1
10
=
1 )3)(2(
)5(
k kk
kk J1061
.10000978265649614187... Likk
k10 10
1
1
10
1
1010 0
1
10
, ,
.10015243644321545885... | ( )|( ) k k
kk
1
91
.10017140859663285712... ( )3
12
.10031730167435392116... 3
2 12
1 1
2
11
12
32
2
( )
logk
k k kk k k
.1004753... 1
2
1 a non trivial
egerpower
int
...
1
0
1
0
1
0222
2222
14
16
3dzdydx
zyx
zyx
... 3
10
.100637753119871153799... Likk
k4 4
1
1
10
1
104 0
1
10
, ,
... 8 2
3
4 0
1
G
x x xarctan log dx
... 3 2 2
16 1 30
1
log log
( )
x
xdx
... 2 2
3
13
36
1
1 3
1
1
log ( )
( )(
)
k
k k k k
.1010205144336438036... 5 2 62 1
2 3 11
( )!!
( )! ( )
k
k kkk
1 .1010205144336438036... 6 2 61
12 1
2
0
kk k
k
k( )
... ( )3 1
2
.10109738718799412444... 1 2 2 2 2 6 2 6 13 2 3
0
1
log log log log ( ) x dx
... k kk k
kk k
2
1
4 4
4 32
4 11
1 ( )
( )
( )
... K3 1( )
... Likk
k3 3
1
1
10
1
10
1
103 0
, ,
... 2
)2(
2
log)1( 2
22
k
k
k
k
... 2
... ( )
( )log
1
2 1 3
3
3
4
32
1
1
k
kk k k
... 1
11 11
3
1131
1
1k
k
k
kk
k
( )( )
... 1
10
2 5
55
1
520
csch ( )k
k k
... log ( ) 22
kk
... k
kk !
1
.10177395490857989056... G
k kk9
1
12 9
1
12 32 21
( ) ( )
= x dx
e ex x3 30
.10187654235021826540... 1
2
1
3
1
2 2 1
1
0
arctan( )
( )
1
2
k
kk
k
k
... 8
9
20
27
4
3 4
2
1
logk Hk
kk
... ( )k
kk 21
.102040816326530612245... =5
49 7 11
( )kk
k
... e e
e
e
k k
e
e
k k
k
1 1
220
( )
!( )
.1021331870465286303... 75161
3 2
3 3
0
e k
k kk ( )!3
.10218644100153632824... 28
2
14
5
84
1
2 1 2 80
log ( )
( )(
k
)k k k
.10228427314668489686... 1 2
3
1
3 607 4
1
log ( )k
k k
dx
x x
1 .10228427314668489686... 4 2
33
0
2
logsin( ) log tan
/
x x
dx
... 1
2
2 1 2
21
2
0
1
cos log sin log sin
e
x x
xdx
x x
edx
e
x
... 1
101 ( )k kk
.10261779109939113111... Likk
k2 2
1
1
10
1
10
1
102 0
, ,
=37
360
1
6
1
21
( )k
k k k
= x x dxx
dx
x5
0
1
171 1
1log( ) log
.10280837917801415228... 2
2196
2
16
1
4
( )
( )kk
1 .10317951782010706548... ( )4 1
2 214
1
k k
kkk k 1
1 .10322736392868822057... 22
11113
22 2
1
231
k
k k
kk
HypPFQ
, , , , , , ,
2 .10329797038480241946... 5
55 3 13
2 1kk
k
k
k
( )
.10330865572984108688... 106
158 2
1
2
1
2 2 70
arctan( )
( )
k
kk k
1 .10358810559827853091... 1 21
log ( ) kk
.10359958052928999945...
( )
( )log3
42 2 3 2
2
4 21
2
31
x
x xdx
x dx
e ex x
= x x
xdx
2 2
20
1
1
log
GR 4.262.13
2 .10359958052928999945... 7 3
42 3 2
1
2 1 30
( )( )
( )
kk
= log2
20
1
1
x
xdx
GR 4.261.13
= x dx
e ex x
2
1
.10363832351432696479... 3
1 11
11
12
1
5
0e
k
k
k
kk
k
k
k
( )( )!
( )( )!
1 .10363832351432696479... 3
12 1
1
2
1e
k
kk
k
( )( )
( )!
.10365376431822690417...
2 1 20e
ex
xdxe
cos
AS 4.3.146
.10367765131532296838... 10
9
1
4
2 1
2
1
2 4
2
1
( )!!
( )!!
k
k kk
.10370336342207529206...
2
2
22
2
2 8
22
2loglog
log
4
= x e x dxx2 2
0
2
log
.10387370626474633199... 11
24
2 2 2
16
1
442
cot coth
kk
=
1
1 1)4(4k
k k
.1039321458392100935... 5 8 2 2 2
41
3
21
1
log log( )k k
k
H
k
... G k
k
k
k4
1
8
1
4 12 21
( )
( )
.10403647111890759328... dx
x x6 31 1
.10413006886308670891... 3
64 2 22 30
dx
x( )
1 .10421860013157296289... 1
3 3 10k
k k( )
... 2
21
1
3242 1
1
4 2 11
2
log( )
( )( )
k k
k
k
kk
k
... log !10
... 2
2 2
2
21sinh
k
kk
... 2 8
2
4
3
4
2
0
log log sinx x x
edxx
...
3 3
1
2
1
2 215 4
1
k k
k
dx
x x xk
! ( )!
( )! 3
.1046120855592865083... si x x x dx( ) sin log sin1 10
1
... 2
1
22
1
41
2 1 11 1
12
12
1
Fk k
k kk
, ,( )!( )!
( )!(
)!
... 8
9
9
81
81
1
log ( )
k
k
kk
H
1 .1047379393598043171... 4
1 11
2 21
erf erfix
dx
x
cosh
= 3
14
1cosh2
x
dx
x
... eee
... ( ) ( ) )( )
( )
1 3 14 1
11
1
33 6 3
3 42
k
k k
k kk k k
k
... 617
5880
1
1 71
k k kk ( )( )
... 71 431 2 3
2
1
ek
k kkk ( )! ( )
... ( )
( )
k
kk Li
kk k
1
113
23
1
... log( )
log
2 2
2
161
1
1612
1
k
k kkk k
=
1
1 10
1 2
2
x
x
x
x
dx
xlog GR 4.267.5
... 11
121
k kk ( )
... 7
42
1 1
14 22
22
( )( )k k k kk k
= ( ) ( ) ( )2 1 1 22
1
1
k k kk
k
k
1
=
rpoweintegertrivialnonaω
2 1
1
... 13
8
...
( )
!
k
kk
1
2
21
... 5
8
3 2
4
1
2 122
log
( )kk k
... 10 ... e1 10/
... sin ( )
( )
1
8
1
2 1
1 3
1
!
k
k
k
k
.1052150134619642098...
ci x x dx
2 0
2
sin log/
=2
19
... log log , ,10 91
10
1
101 0
1
kk k
= 21
19
1
19 2 12 10
arctanh
kk k( )
1 .1053671693677152639... 1
2 1 222
11 ( )
( )k k
kk k
k
... ( ) ( ) ( ) ( )( )
( )3 2 3 2 5
2
1 41
k k H
kk
k
.1055728090000841214... 12
5
1 2 1
2 4
1
1
( ) ( )!
( )!
k
kk
k
k
!
... ( )3 1
2 213
2
k k
kk 1k k
.10569608377461678485...
0
3 log4
1 2
dxxex x
... ii i i
k i k ik61 1
1 13 34 4
1
( ) ( )( ) ( )( ) ( )
.10590811817894415833... 2 3
32 2 3 2
2 2
332 2log
log log loglog
log
= ( )
( )
1
2 1
1
1
kk
kk
kH
k
.10592205557311457100... 4 28
3
1
2 2 61
log( )
kk k
1 .10602621952710291551... 4
41
1
16 21
sinh
kk
.10610329539459689051... 1
3
.10612203480430171906... 10
2
5
26
751
12
16
log
logx
dx
x
... 114
2 4k ks d
k log( )
s
...
4 42 20e
x
xdx
cos
= GR 3.716.4 cos( tan ) sin/
2 2
0
2
x x d
x
... k k
k
2 11
2
( )
.10634708332087194237... log log2
2
2
2
2
=( )
1
11
kk
k
kH
k
.10640018741773440887... loglog( )2
2
20
1
212
1
2
1
3
Li
x
x
.10642015279284592346... 1
82 2 2 21 1 2 2i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) i
= ( ) ( )( )
( )
1 2 1 11
11 2
1
2
2 32
k
k k
k kk k
k
... 24 5 14 32
52 1
1 11 11
1
4 2
52
( ) ( ) ( )( )
k k k
k
3
k
= ( ) ( )
1 14
1
k
k
k k 1
=8
754
0
1
x xarccos dx
... 1
6
3
12
3
2
1
4 3
1
21
csch
( )k
k k
... ( )( )
arctan
11
2 1
11
1
32 1
k
k k
k
kk
k k
... 1
10
10
20
5
2
1
2 520
csch( )k
k k
... 12
2)1(
5
2arcsin2arctan
12
0
k
k
k
k
.10730091830127584519... 1
2 8
1
4 1
1
2 21
( )
( )
k
k k GR 0.237.2
= 1
16 1
2
1621 1k
k
kk
k
( ) GR 0.232.1
= ( )
1
4 60
k
k k
= 1
7 31 x x
1 .10736702429642214912... 3
28
.10742579474316097693... 1 44
5
1
4 1
1
1
log( )
( )
k
kk k
... one-ninth constant
.1076723757992311189... 10 62 21
ek
k kk ( )!
... e Ik
kk
0
12
20
1
2
( )
( !)
!
...
4 2
1
21 1
21
2coth
i i
= k
k kk k
k
k
kk
1
4
1
42 23
1
1
1
( )( ) ( ) 1
...
0
2
)!3(
2)1(
4
1sinh
8
1
k
kk
k
ee
.10819766216224657297... 3
2
3
2
1
2
1
2 22
log( )
( )
k
kk k k
.1082531754730548308... 3
16
1
12
2
0
( )k
kk
k k
k
... 432
9
1
3 2
1
21
Gk
k
k
( )
( / )
... 1
4 11k
k k ( )
1
1 .10854136010382960099... Hkk
k 2 11
...
161
14
2 2
0
2
ee x xx sin cos dx
... ( ( ) ) k
kk
1 2
22
... 27 4 3
481
13
15
logx
dx
x
.109096051791331928583... 71 6
108
1
31
2
21
2
0
1
k kx x
k ( )log( ) log x dx
...
12 )22(4
)2(
4
)3(7
4
1
2
log
kk kk
k
... x dx
x
6
120
1
1
=7
641
71
1
( ) kk
k
k
... 1
2
1
21
1 2
k
ke
k !sin sin sincos( / )
... 1
12
2
42
1
2 3
1
21
csch( )k
k k k
... 4 2 5
6
1
4 2
21
1
( )
( )
k
kk k
k
k
... ( ( ) ) k
kk
1
2
2
2
... 1
212
0
12
Ei e Ei e dxe x
( ) ( )
2 .1097428012368919745... 1
22 k kk log
=9
82
1
2 91 2 2 1 3
0
cosh log( ) ( ) logk k
k
e J943
.10981384722661197608... log( )
27
12
1
506 5
1
k
k k
dx
x x
...
4
2
2 1
2
20
4
arctan sin
sin
/ x
xdx
... 1
132 k
k
kk
log
... dxx
xx
1
032 )1(
arcsin
64223
... 3 3
4 122 2 3
12
4 32
( )log
( )
k
k k k
9 .1104335791442988819... 83
...
( )
( )
( )3
4 41
k
kk
Titchmarsh 1.2.13
...
2 21
2 1
2 2 2 1
2 1
8 2 11 1
( )!!
( )!! ( ) ( )
k
k k
k
k kkk
kk
=( ) ( )
1
4 3
1
4 1
1 1
1
k k
k k k J76, K ex. 108c
=
0
2/
12
)1(
k
k
k Prud. 5.1.4.3
= 11
2 30
( )k
k k
= dx
x40 1
Seaborn p. 136
=dx
x
dx
x
x dx
x20
20
2
402 2 1
1
= dx
x x( )1 12 20
1
= d
1 20
2
sin
/
.11074505351367928181... log( )
log2
3 3 31
1
3
1
12
k
k kkkk k
... ( )
1
430
k
k k
... 1
10 11
3
1031
1
1k
k
k
kk
k
( )( )
... 9 4 6 3 265
16 3
2
41
( ) ( ) ( )( )
k
kk
... 4
... log3 2
3
... ( )
!
k
k kk3
2
... 2
3
... e Lie e kk
k2 2
0
1 1
1
( )
=1
9 41 0
2
2
log
log(sin ) sin cos/x
xx x x
dx
= ( ) ( ) ( )k k
kk
kk
kk3 1 6 1 91 1 1
k
1
.1114472084426055567...
1
21
1
21
1
21
=1
21
2 21
1
2
cot
=1
2
2 1
232 1k k
k 1
kk
k
( )
1 .1114472084426055567...
2
11
2
11
2
1
=1
2
2 1
231 1k k
k
kk
k
( )
.11145269365447544048...
2 3 2
36
2
3
2
32
1
2
3
23
log log( )Li
=log
( )( )
2
21 2 1
x
x x
... 1
2
5
4
1
9 2 12 10
log( )
arctanh1
9 kk k
= Im logii
12
... 1
541 kk
1 .1116439382240662949...
2
3
2
1
2
3
2
13333
333
1 iiii
i
= 1 1
21 3 12 1
1
1
1k kk
k
k
k
( ) ( ) ) 1
= 1 1 3 3 11
1
( ) ( ) ( ))k
k
k k
... ( )
1
5 607 2
1
k
k k
dx
x x
... Li k kk
41
1
9
1
9 4
... ( )kk
15
2
... 2
1
2 1
23 2 22 1k k
k
k
1
k k( )
( )
.11207559668468037943... ( ) ( )
( )
2
32
2
16
16
16 11
2
2 21
k k k
kkk k
.11210169134154575458...
1
21 1
1
1 2 21
i i
k kk ( )
... ( )kk
k 2 111
... e
F
k
kk
1
1
/
...
1
2
232 )(
)4(
)2(
12
5
k k
kd
Titchmarsh 1.2.9
=
p p
p22
2
)1(
1
... 8
2 2
1 1
2
12 9
15
1e
e
x
dx
x x
dx
x
sinh sinh
... 1
2421 3 2 22 csc
= 1
48 2 22 3 2 21 22csc sec sin sin
=( )
1
24 22
k
k k k
... 21
41
/k
k k
... 3 3
32
1
2 2 30
( ) ( )
( )
k
k k
=
1
2
1
23 31
Li i Li ik
k
k
( ) ( ) cos
=
1
0
1
0
1
02221
dzdydxzyx
xyz
... Likk
k3 3
1
1
9
1
9
...
( )
( )
k
kk
2 1
2
... 1
22 1 1
1 1
2
2
0
1
log( )ee
e
edx
x
x
... 1 1
11
, ,
kk
k
... 3
43
4
3
1
3 1
1
1
log( )
( )
k
kk
k
k
1 .11318390185276958041... 1
18 7 6 5 4 3 20 k k k k k k k kk
.11319164174034262221...
log
4
3
.11324478557577296923... ( )jkkj
4 111
4 .11325037878292751717... e 2
.11356645044528407969... 5
361
5
61
51
1
log ( )k kk
k
H k
.11357028943968722004... ( )
11
2
k
kk k
.11360253356370630128... log
!
k
k kk 22
.11370563888010938117... 3
22 2
1
4 12 21
log( )k kk
J104, K ex. 110e
= 1
2 1 2 2
1
2 11 1k k k k k 2kk
k( )( ) ( )(
)
= k
k k
k k
kk
k
1
4
2 2
431 1
1 ( ) ( )
= ( )
12
2
k
k k k
= dx
x x
dx
e ex x( ) (2 21
21 1
)
.1138888888888986028...
| ( )|( ) k k
kk
1
9 11
.11392894125692285447... 616 1
40
3
21
e k k
x
xdx
k
k
e( )
!( )
log
.11422322878784232903...
9
112
7 2
2242 2
3 43 4 3 4
// /csc
csch
= ( )
1
842
k
k k
.11427266790258497564... 4 8 2 8 2 3 2 3 3 3 12log log log log log
= H
kk
kk 4 21 ( )
... Likk
k2 2
1
1
9
1
9
= 3log33
16 2
2
2
Li
... Ei Eik kk
( ) ( )( )!( )
1 1 21
2 1 2 10
=
2 2
1
2
3
2
3
2
1
4i si i HypPFQ( ) , , ,
=11
96
1
2 41
k k kk ( )( )
.11466451051968119045... 4 2 4 22
32
2 22
2
21
log log( )
k
kkk
.11467552038971033281...
1
24 2 6 2 66 6
2 6csch sech
sinh
= ( )
1
6 1
1
21
k
k k
.11468222811783944732... 2 2 5 4 21 2 1
2 4
1
1
log log( ) ( )!!
( )!
k
kk
k
k k
... 1
16
1
2
1
8
1
2
1
2 4
1 2
1
cos sin( )
( )!
k
kk
k
k
... 2 1tan
... 1
2
11
1
Ei e Eie
k
k kk
( )sinh
!
... J2 1( )
...
23 1
)1(
k
k
k
... 1
10
3
5
4
54
6
5 5
cot cot cos logsin
=
1
104
8
5 54
12
5
2
5cos logsin cos logsin
=
1
104
16
5
2
5cos logsin
=1
5 1 5 21 ( )(k k )k
.11524253454462680448... dxex
x
e
e
1
02)(
log1)1log(
... 5
... log
log log (log )( )3
22 3
2
3
1
22 3 2
1
2
1
2
7 3
8
Li Li
=log ( )
( )
log2
0
1 2
21
21
2 1
x
x x
x
xdx
... log
( )( )( )
2
6 1 20
dx
x x x 4
... 3
256
2
...
log log log2 63
2
2 3
2 31
16 2
1 x
dx
x
=
126
1
0
6 11log)1log(
x
dx
xdxx
... 1
2
1 5
2
1
2
2
2 1
1
2 22
2 20
log( ) ( )!!
( )!!
k
kk
k
k k J143
.11591190225020152905... 1
4
1
2
1
2 2 12 11
arctan( )
( )
k
kk k
.11593151565841244881...
11 1
1)12(
2
)(2log1
kkk k
kk
=
1 )12(4
)12(
kk k
k
=
kk kk
log 11 1
22
= cos xx
dx
x
1
1 20
... j3 J311
... 2
312
9
4
27
12
3
sin k
kk
= iLi e Li ei
2 33
33( ) ( ) i GR 1.443.5
1 .1165422623587819774... 1
3
1
4 8
4
40
cossin x
xdx
... 7 3
4
2
2
1
4
2
4 2 221
( ) log ( )
( )
k
kkk
... arcsinh
2 2
2 1
2 31
2
2 2 11
1
( )( )
kk
k
k
k
k
k
... 7
8
8
7
1
7
1
1
log( )
k
kk
k
H
... 2
21
2 2116
1
2
1
2 1 2 1
1
4 1
(( )( )) ( )k k kk k
J247, J373
... ( ) log
( )3 24 2
5
332
1
2 1
2
3 21
k kk
=15
128
1
3
4
1
( )k
kk
k
...
12
13
90
1
2 1 2 70
( )
( )(
k
)k k k
... 2
3
1
2
1
4 2 10
arctanhk
k k( )
... 1
12
3
2
3
23 8 3csc sec sin
=( )
1
4 1
1
21
k
k k k
... 2 1
2
1
3
2 1
2
1
2 3
2
1
G k
k kk
( )!!
( )!! J385
...
4
2112
4 2 2
( ) ( )( )
Lr k
kk
2 .11744314664488689721... 2 24 1
1
4( ) /
k
k
.117571778435605270... ( ) ( )
( )
( )3 2
4
43
2 21
H
kk
k
=2
17
... log( )
, ,9
8
1
9
1
8
1
91 0
1
1
1
kk
k
kkk k
= 21
172
1
17 2 12 10
arctanh
kk k( )
K148
...
1
1)2()2)(1(
2
1
k
kk
kk
... dxx
xdxxx
4/
0
34/
0
2
tan
sincossin
26
1
1 .1180339887498948482... 5
2
1
2
2 1
200
k
k kk
.1182267093239637759... 3
21 1
1
2 2 21
cos sin( )
( )!( )
k
k k k
... 11
21 1 1 1 2 2 ( ) sin (cos ) log( cos ) 1
=cos
( )
k
k kk
11
... G k
k
k
kk
log ( ) ( )
( )
2
4
4 1 2
16 2 11
= 21
22
1
41
kk kk
arctanh arctanh
.118333333333333333333 = 71
600
1
1 61
k k kk ( )( )
... 1 2
1 k
k
kk
log
.118379103687155739697... 7
5760
4
4 4
Li i Li i( ) ( )
...
3
1
6
5
432
1
36
)3(13
3162
5 )2()2(3
=
03)23(
)1(
k
k
k
... 1 11
e x ee x, dx
.11862641298045697477... 1 1 2 1 11 2 1
2 2 1
1
1
log( ) ( )!!
( )! ( )arcsinh
k
k
k
k k J389
= ( )
( )
1
4 2 1
21
1
k
kk k
k
k
.11865060567398120259...
H kk
kkk k
1
2
2
22
2 11
1( ( ) )
log
... 2 3 11
3
30
e Eik
k kk
( )!( )
... 2
8
1
4 12
20
1
log
( )
x
x xdx GR 4.241.11
...
4
2
3
1
2 506 4
1
4
0
4
( )
tan/k
k k
dx
x xx dx
... 8
1
22
1
2 1
2
0
1
logarctanx x
xdx
... k
kk
2
0 1( ) !!
... ( )
!
k
k
!!
k
1
0
.1189394022668291491...
4
33
4 31
3
2015
51
83
1
( ) log x
x xdx
x
e edxx x
.11910825533882119347...
5
36 144
2
18
2
0
1loglogx arccot2 x x dx
... 2 1 1
2
2
21
sin log coslog
e
x x
xdx
e
... 1
1
12
1
21 0e e
ex
edx
k
kk
x
( ) cos( )
... ( ) ( )4 5 2
... 3
8
.1193689967602449311... ( )( )
12 1
1 1
1
kk
k
k
... 4 4
.1194258969062993801... 2
21
8
144
1
12 9
G
kk ( )
... cossin ( )
( )1
1
2
1
2 2
1
1
!
k
k
k
k
... ( ) ( )( )
3 41 4
1
k
kk
... 1
4
1
2
1
2 4
1
1
sin( )
( )!
k
kk
k
k
... Ei
e
k
k
k
k
( ) ( ) ( )
!
1 1 1
1
1
=3
25 5 11
( )kk
k
.12025202884329818022... 1
2
1
2
1
2
4
5
1
2 22 11
arctan log( )
( )
1
k
kk k k
... 7 3 3
15
2 1
2 31
1
k
k kk
k
k
( )
( )
... ( )
( ) (3
12
1
2 1
2
21
)2
H
k k kk
k
... ( )
( !){},{ , ,},
11 1 1
30
k
k kHypPFQ
... e e ee
e
k
e k
k
kk
log( )( )
( )
11
1
10
...
2)1(coth
2122
1
i
k
H
k
k
= i
i i i4
1 1 1 12 2 1 1 ( ) ( ) ( ) ( ) i
... ( log )
arcsin log1 2
8 0
1 x x x dx
... 2 2
5
47
300
1
1 5
1
50
1
21
log
( )( )
( )
k k k kk
k
k
= log log( )11
11
64
0
1
x
dx
xx x dx
... 3
21
230
log( )k
k k
...
4 2 2 2
4 1 2
81
tan( ) (
)
k
kk
k
=1
2 1
1
8 1
6
2 1 8 12 21
2
2 21k k
k
k kk k
( )( )
.12078223763524522235...
log log log
2
1
2
4 3
2
=
1
2
1
1 20 e
dx
e xx x GR 3.427.3
=0
2
2
sinh
cosh
x dx
xe xx GR 3.553.2
.1208515214587351411... 3 3
212 2 10
1
1
1
3 31
( )log
( )
( )
k
k k k
.12092539655805535367... 1
642 kk
.12111826828242117256... 3
30256
1
4 2
( )
( )
k
k k
...
( )
1
2
1 2
1 22
1
k
kk
k
kk
... 1
6
1
2 2
14
2
2 2 21
2
coth
( )k kk
... log5
6
.1213203435596425732... 3
22
1
4 1
21
0
( )
( )
k
kk
k
k
k
k
= sin tan/
x x d2
0
4
x
... ( )2 1
4 11
kk
k
...
3
1
9
1
29
)2(4 )1(2 g
=log logx
xdx
x x
xdx3
0
1
311 1
... ( )3
2
... 43
2
3
2
1
2
1
21 3
1
2 2 60
log , ,( )
( )
k
kk k
... 14
153
41
( )( )
a k
kk
Titchmarsh 1.2.13
.12202746647028717052... 3
66 2
9 3
21
1
2 30
loglog
log( )x
dx
... 1
8 42
1
4
1
21
csch( )k
k k
=sin cosx x
edxx
10
...
03
31
)2(3
)(
12
3
1
9
34)3(26
k k
.122239373141852827246... ( ) ( )
1 3
9
1
9 1
1
13
1
k
kk k
k
k
1 .12229100107160344571... ( ) ( )
( )4 5 1
51
k
kk
HW Thm. 290
=
14
1
( )k
kk
.12235085376504869019... 12
21
2 22
log
( )k
k k k
2 .12240935830266022225... ( )k
k kk
22
.12241743810962728388... 21
4
1
2 42
1
1
sin( )
( )!
k
kk k
.12244897959183673469... 6
49
1
6
1
1
( )k
kk
k
1 .122462048309372981433... 2 11
6 11 6
1
1
/ ( )
k
k k
.12248933156343207709... 1
2
1
4
1
8
1
16 2 103 1
2
arctan( )
( )
k
kk k
dx
x x
... 2
2
0
18 2 11
361
1
loglog logx
xx dx
=log( ) logx x
xdx
14
1
.1228573089942169826... 7
21
1 2
1 2
0e
k
k
k
kk
( )
( )!
.12287969597930143319... loglog log ( ) log ( )
( )2 1
2
2
2
3
3
4
1
1
2 3 2
20
1
x
x xdx
.1229028434255558374... sin ( )
!( )
2 2
2
1 21
20
k k
k k k
... 325
48
15 1
12
51
( )
( )
k
k k
=8
65
1
2 81
1
3 2 1 2
cosh log
( ) ( ) logk
k
ke J943
4 .12310562561766054982... 17
1 .12316448278630278663... 3
2
1
3
1
3 2 10
erfi
k kkk
! ( )
2 .12321607812022000506...
1
221 )12(
2
2
)12(
2
1
2
1
24
1
kkk k
kkk
... Champernowne number, integers written in sequence
.123456790123456790 = 10
81 10
1
21
4
1
k kk
k
k
kk
( )
... 37
226803
62 ( ) 18 MI 4, p. 15
= 2
36
1
32 4
1
32 2
13 2
2
41
( ) ( ) ( ) ( ) ( )( )
H
kk
k
.12360922914430632778... 1
3
2
9 121 1
k
kk
k
k
( )
... 21
22
1
2 4
1
1
cik k
k
kk
log( )
( )! ( )
... 355
1082 4 3
4 31
( ) ( )( )
k
kk
8 .1240384046359603605... 66
... 3
2
4
3
1
3 12
1
1
log( )
( )
k
kk
k
H
k
...
2
1
41
21
2
1
2 4 121
i i
k kk ( )
=( ) ( )
1 2
2
1
2 11
k
k
1
k
k
529 .124242424242424242424 = 174611
330 20 B
...
2 41
1
4 coth ( ) ( )i i
=1
1
1 1
13 22
3 12
42k k k k k kk k k
... 4
251
4
5
1
4
1
1
log( )k
kk
k
kH
... 5
2
5
4 52
1
log( )
1
H
kk
kk
.124511599540703343791... ( )( )
18 1
1
1
kk
k
k
... 1
9 32
1
6 4 6 11
log( )(k kk )
J263
... 11 215
2 2 10
log( )(
3)
k
k kkk
... 8 22
312
1
2 1
2
2 21
log( )
k kk
... 1
512
1
163
1
4 1
2
40
1
, ,log x
xdx
6
... ( )
1
2 1
1
1
k
kk
k
... sinh sin cos( )
cosh( )/ /2
4
2
4 2
1
2
1
2
1 4 1 4
i
=sin /
( )!
k
k kk
2
2 41
.12500000000000000000 = 1
8 4 21
1 2
k
kk ( ) J364
=
1 )3)(2)(1(k
k
kkk
H
=
1
5
)1(kkke
k
= ( )k
kk 4 11
= x x x3
0
1
arcsin arccos dx
= log 11
21
5
x
dx
x
1 .125000000000000000000 = 9
83
2 H ( )
4 .125000000000000000000 = 33
8 3
3
1
kk
k
... 1 2 12 4 6 3 8 52
3
51
( ) ( ) ( ) ( )( )
k
kk
.125116312213159864814...
2
2 21
26
152
2
4
1 1
( )( )
( )
( ( ) )k
k nk n not squarefree
.12519689192003630759... 1
6
4
9 3 2
1
2
2
20
2
( sin )
( sin )
/ x
xdx
...
1
8192
1
8
1
16 22
30
( )
( )kk
... ( )
( )! (
1
1 2 2
1
1
k
kk k k )
Titchmarsh 14.32.1
1 .12522166627417648991... dxx
x
0
4
442
sinh)4()2(2
453
1 .1253860830832697192... 12 1
10
1
eEi Ei
e dx
x
x
( ) ( )( )
2 .12538708076642786114...
1
22
2
22log
6 kkkH
...
1
3456
1
6
1
12 102
31
( )
( )kk
... csc ( )3
6
1
18
1
9
1
2 21
k
k k
.12549268916822826564... 3
6
5
3
1
12
7
3
3
18 3 32
arctan log
dx
x x
... ( )k
kLi
k kk k4
24
1
1 1
. 1255356315950551333...
2 2
int2
11
)(11)()(1
k npowereger
nontrivialak
k n
kkk
=
S )1(
1, where S is the set of all non-trivial integer powers
... 1
22 2
7
2
1
4
1
33 3
5
2
1
42 1 2 1F F, , , , , ,
=
1
21
1
21
2)1(
)!2(
!!)1(2arccsch
125
54
25
4
k
k
k
k
k
kk
k
kkk
... 7
2 15 9459 3 3 5 28
1
1
2 4 6
6 31
( ) ( )( )k kk
... ( )k
kLi
k kkk k2
1
2
1
222
21
... 2
92 1
2
2 21
sinh( )
kk
...
1
1458
2
9
1
9 72
31
( )
( )kk
... Likk
k4 4
1
1
8
1
8
.12613687638920210969... dx
x x3 22
... 215
2 305 3 5
1
1
2 4
5 31
( ) ( )( )k kk
...
3
23
1512
7 3
128
1
1024
1
4
1
8 6
( )
( )( )
kk
...
3
2
2
31
3
1
1
13
20
1
log loglog
x x x
dx
... 1
2 3 3
2
3
1
3 506 3
1
log ( )k
k k
dx
x xdx
... 1 3 2 4 3 1 12
2
( ) ( ) ( ) ( ) ( )k
k
k k k
=4 2
1
3 2
2 31
k k k
k k
1
k
( )
... 11 3 1
6 2 1 12 110
dx
x /
... 7
2
... 11 4 e
... log 1 2 e
.1269531324505805969... 1
2
3
8 131 1
k
kk
k
k
( )
1 .12700904300677231423...
4
34
3 3
4
3 3
11
3 1
2
21
i i kk ( )
... Likk
k3 3
1
1
8
1
8
.1271613037212348273... 3 2
4 8
1
16 4 421 2
log ( )
k k
k
kk
k
= dx
x x x x4 3 21
.12732395447351626862... 2
5
3 .12733564569914135311... 2 22 1
1
2( ) /
k
k
.127494719314267653131...
( )2 12
1
k
kkk
.12755009587421398544... ( ) ( ) ( )( )
4 10 2 2 3 151
14 31
k kk
.12759750555417038785... 13 3
216
4
1296 3
1
432
1
3
1
6 4
32
31
( )
( )( )
kk
1 .12762596520638078523... cosh( )!
/ /1
2 2
1
2 4
1 2 1 2
0
e e
k kk
AS 4.5.63
= 11
2 120
( )kk2 J1079
.1276340777968094125... ( )
1
2 111
k
kk
...
( )
( )
4 1
2 1
... 4 26
1
2
1
2
2
3 21 1
log( )
k k
k
kk
k
=2
1
0
22 )(
x
dxxLi
... 21
4
1 1
2 42
1
arcsin( )!(
( )!
k k
k kk
)! K ex. 123
... 1
4
5
3 142
log
x dx
x
... 120
1
( )kk
k
... 1
4
2
4
3
12 1 2 32 2 20
dx
x x x( )( )( )
... k kk
kk
k
3
1
3
12 1 2
( )
... 2
1 ... sin ( ) cos ( ) ( ) sin1 1 1 1 1ci si
= x x x dlog sin( )10
1
x
1 .12852792472431008541... k
kk
k
k
k
2
41
1
11
1
21 4 2
( ) ( ) 1
... 2 2
3
1
31
13
14
loglog
x
dx
x
... 5
6
... 1
2 21
k kk
( )
... Jk k
Fk
k3
00 12
1
34 1( )
( )
!( )!; ;
LY 6.117
.12895651449431342780... 1
250
2
5
1
5 22
30
( )
( )
kk
.12897955148524914959... 1
81 ( )k kk
... 888
5 51
2 1
161
1
( )kk
k
k
k
k
.12913986010995340567... 1
8 21
kk k
... 12 1
1 k kk
... 27
23 1 3
1
9 5 31
log ( )
k kk
1 .12928728700635855478... 1
17 6 5 4 3 20 k k k k k k kk
... 2 1
2112
21
1
log( )
( )
k
k
k
k J347
= k
k k
k k
kk
k
1
4 2
1
23 21 2
( ) ( )
=log
( )
log
( )
log
( )
2
31
2
30
1
20
1
1 1
x
xdx
x x
xdx
x x
xdx
1
.12950217839233689628...
ee
k
k
k
1
41
/
.12958986973548106716... ( )3
12
1
4
2 12
6 5 42
3
k
k k k kk
= k k kk
( ) ( )2 22
1
2 .12970254898330641813... 1
1 1 13
0 ( !){},{ , },
kHypPFQ
k
3 .12988103563175856528...
iii
2
1
2
1tanh
.12988213255715796275... 13 33 2
2 1k k kk k
k k
( ) ( )1
.12994946687227935132... 1
6 CFG A28
1 .13004260498008966987...
Gx
x xdx
2 8
7 3
42
1 1
2 2
20
1( )log
arccos
( )( )
.13010096960687982790... 18
7 3
4
2 1
2 3
2
30
( )
( )
k
kk
... Hk
kk
3
1 3
.13027382637343684041... 1
4
1
2 20
sinh( )!
4
k
k kk
... 1
22 1
12
2
log( )
k
k kk
= 11
21
1
2
k
kk
kk
( ) log
.13039559891064138117... 101
12
31
3
k kk ( )
K ex. 127
... ( )
( ) ( )( )
k
k k k kmk m
kj j
jk mk
1
1
11
2 22 21
1
... 02 12
1 1( ) ( ) ( )k k jkk kj
=1
1
1
1 122 2k
k
k k
k
kjjk k k
( )
( )
( )
2
... 02
1( ) ( )k kk
...
4 32
2
2
22
0
4
log
tan/
x x dx GR 3.839.1
... log
( )
( ( )) logk
k k
k k
kk k2 2 2
22
21
1
=j k
k jkj
log2
21
... 3
4
1
4
1
20
e
e
k
kk
( )!
... ( ) ( )
log
1
2
1
3
1
21
1
2
2
2
k
k k
k
k kk k
k
... 1
8 22
1
420
csch( )k
k k
... cos cosh
( )
/ / 1 4 1 4
02 4
!
k
k k
... 2
2
0
1
44 6 G xarcsin log x dx
... i i( ) cos ( ) ( ) cos ( )/ / /1 1 1 13 4 1 4 1 4 3 4 /
... H
kk
kk 8 2
1
... 1
1 k kkk !
... 1
22 22 2 ( ) ( )( ) ( )i i
=x x
e edxx x
2
0 1
cos
( )
... 7 3
64
1
4 3 31
( )
( )
kk
... ( )( )
11
2
3k
k kk
4 .1315925305995249344... 1
22
( )kk
... log
arctan7
6
3
6
3
3
5
3 132
dx
x
... k
k k k k k ks d
k k
1 1 1
142
3 42 3
4
log log log( ) ) s
...
2
3 31 2
414
41
384
3
4
1
4
1
1 2
Gk
k
k
k
( ) ( ) ( )
( / )
... 4
2
21
1
2 2 2 30
log ( )
( )(
k
)k k k
= 12
( )kk
=
134
2/1
0
2 11log)41log(
x
dx
xdxx
... 4
2
2
1
2 2 1
1
1
log ( )
( )
k
k k k J154
=
1
2/
1
)1(
k
k
k Prud. 5.1.2.5
= xx
dxlog 11
40
1
= log( )
11
2 10
x xdx
=
12
arctandx
x
x
.13197325257243247...
ieLi e e Li e
k
k
ii i i
k2 132
3 31
( ) ( )sin
( )
3 .13203378002080632299...
3 3
2 2 3
1
3
13
13
log ( )
( )
GR 8.336.3
.1321188648648523596...
1
1
2)1(
3
2log1
9
2
kk
kk kH
.1321205588285576784... 1
2
1 1
2
1
1
e k
k
k
( )
( )!
... 6
7
7
6
1
6
1
1
log( )
k
kk
k
H
... 1
742 kk
.13225425564190408112...
( )5
21
kk
k
11 .13235425497340275579... 5815
729 33
6
0
ek
k kk
!
.13242435197214638289... log( ) 2
.13250071473061817597... 3
21
2 2
7
1
14
1
3 3 20
log ( )
( )(
k
)k k k
25 .13274122871834590770... 8
1 .132843018043786287416... ( )( )
12 1
1
1
kk
k
k
.13290111441703984606... 1 1
2 21
2 22 2 1
0e eek k
k
cosh( ) ( ) J943
.13301594051492813654... 2 24
3 2
23
1
2 1
2 2
2 21
loglog
( )
k
k k k
.13302701266008896243... 116 8
5
8
2
4 8
3
8
cot cot logsin logsin
= ( )
1
4 506 2
1
k
k k
dx
x x
1 .13309003545679845241...
4 2 2 20log
dx
x x
.13313701469403142513... ( )1 8
1 .13314845306682631683... ek kk
kk
k
8
0 0
1
8
1
2
! !!
.13316890350812216272... 1
2 2 5 5
1
5 1
1
21
csch
( )k
k k
.1332526249619079565... 2 2 2 22 1
2
1
2
log log( )(
H
k kk
k )k
1 .1334789151328136608... 3 5 2 3 41
( ) ( ) ( )
H
kk
k
.13348855233523269955... 5
12
2
32 1 2
2 1
4 2
1
1
log log( )
( )
k
k k k
k
kk
1 .13350690717842253363... ( ) ( )2 2 11
k kk
1
.13353139262452262315... log log8 71
8
1
811
Likk
k
= 21
152
1
15 2 12 11
arctanh
kk k( )
K148
... 3
161
3
4
1
3
1
1
log( )k
kk
k
kH
.13356187812884380949... ( ) log( ) log3
9
1 3
0
1
x x
xdx
6 .13357217028888628968... 2 1
1
k
k k kk
!
...
1
21
1 1
21
1 22
1
( )kk
k
1
=1
12 21 k kk ( )
... 7
2
2
7
2 1
7 2 10
arcsin( )
k
k kkk
.1338487269990037604...
1
0
2sin
10
52sin22cos
5
2xe
dxx
e
.1339745962155613532... 13
2
1 2 1
2 3
1
1
( ) ( )!!
( )!
k
kk
k
k
... 1
1 122 ( )
logk
k
kk
... 1
162 3 33 2 2 2 coth cothcsch csch 8
=1
12 31 ( )kk
... coth
( )
2
4
1
8
3
8 1
1
4
4
4 21
2
e
e kk J951
.134355508461793914859...
03 27327
2
x
dx
... arctan12
2e
dx
e ex x
... ( )
1
3 2
1
1
k
kk
... 1
11
12
1arcsinh
arcsinh2
2
kk
... e e
xdx
x4 1 4
0
... 2
2 2112
11
16
1
1
( )kk
J400
= 1
2
1
12 21
4 22k k k k 2
k k( ) ( )
= ( ) ( )k kk
1 2 12
1 .13500636856055755855... H
k
k
k
k
2
1 2
... cos ( )
( )
1
4
1
2 1
2
1
!
k
k
k
k
...
11
1
3
1
3
1
2
)1(
2
3log
3
1
kk
kk
kEk
LiH
1 .1352918229484613014... dxx
xx
1
0
2
1
log
9
49
3
2
... Ai ( )1
... 2 3 3 3cosh e e
.1353352832366126919... 1 1 2 1 2 1 1 2
20
1
00e k
k
k
k
k
3k k
k
k k k k
kk
( )
!
( ) ( )
!
( ) (
!
)
=( )
( )!
1 2
2
1
1
k k
k
k
k
1 .1353359783187...
( )
( )
k
kk4
2
...
21 2 2 11
1
2 1
cosh( ) ( )
k
k
k k
=cos
cosh
2
0
x
xdx
GR 2.981.3
.135511632809070287634... 1
41
1
4
1
2
1
4 1 2
0
e dx
e
x
x
1 .13563527673789986838... 9
... I2 1( )
... 5
14
2 1sinh
2
1sinh
2
1
4
11cosh
x
dx
x
... k
kk
!
!
10
=163
1200
1
1 51
k k kk ( )( )
... k kk
3 2
1
1 ( )
... 11 182 2
2
1
ek
k kk
( )!
... ii 11sinh2
=
12
1
22 22
)1(
1
)1(
k
k
k
k
kkk
=cos(log )
( )
x
xdx
1 20
1
GR 3.883.1
=cos(log )
( )
x
xdx
1 21
= dxee
xxx
0 )1
sin
.13607105874307714553... 16 23
4
1
41e k
k
k
( )
!( )k
... ( )k
kLi
kk k2
22
2
1
... 2 4 21
2 24
3
0
1
( )log
( )( )
Lix
x xdx
1
.13608276348795433879... 1
3 6
1
8
2
0
( )k
kk
k k
k
... 1
8 11
3
831
1
1k
k
k
k
kk
( )( )
=
4
33
)33(6
33
4
33
)33(6
32
3
2log
3
1
6
i
i
ii
i
... 4 11
1 41 4
1
ek k
k
/
( )!
...
0 0 24
1
4)!2(
!!
4
1arcsin
1515
16
15
16
k k kk
k
kk
kk
... 2 25
4
1 1
1 232 0
log( ) ( )
( )( )(
3)
k
k
k
kk k k k k GR 1.513.7
=
2
1
2
1)1(
)1(2
1
1
1
2k
k
k
kk
Lik
...
2122
)()1(111
1
kk
k
k
k
kk
... 11
1 1k
k k
( ) ( )
... 2 1
2
2 1
2
1
2 2
2
1
( )!!
( )!!
k
k kk
J385
= ( )( )!!
( )!! ( )( )
12 1
2
4 1
2 1 2 21
1
3
k
k
k
k
k
k k J391
.13675623268328324608... 13 3
27
2
81 3
1
54
2
3
1
3 2
32
30
( )
( )( )
kk
... 5
4 2 6
5
2
1
4
2 2
4 21
coth
k
k kk
= ( )( ) ( )
1
42 2
1
k
kk
k k 2
... k
kk
k
k
k
3
51
1
11
1
21 5 3
( ) ( ) 1
... ( )
( )3
26
1
1
2
2 31
k kk
... 1 33
4
1
4 11
log( )k
k k k J149
...
22 1
21
3
3sin
k k
.13707783890401886971... 2
2172
1
6 3
( )kk
=1
12 3
1
12 92 21 ( ) ( )k kk
= log 11
61
x
dx
x
...
( )
( )
( ) log3
32 31
k k
kk
... 9
2
1 .137185713031541409889... 2 2 2
212
2
2
2
28 2
7 3
2 2
log loglog
( )
( )
H
k kk
k 1
...
( ) ( )
( )
3 5
3
... ( )
1
232
k
k k
... 1
2 22 2 1
0sinh( )
e k
k
AS 4.5.62, J942
... 4
5
2
5 5
2
5
4
5 1 5 20
csc /
x dx
x
... ( )
1
4
12
1
kk
kk
F
... e
e xx1 4
02
2/
cosh
dx
12 .13818191912955082028... 2 1 3 1 12
0
log x dx
... log log 11
111
1
ee ek ek
k
= ( )k
e kkk
1
1 .13830174381437745969... I e
eF e
e
k k
k
k
2
0 10
2 1
23
2
; ,!( )!
1 .13838999497166186097... 1
11 k k
k
1 .13842023421981077330... 5
81 3
3
361
1
3 1
1
3 2
1
3 3
3
3 30
( )( )
( ) ( ) ( )k
k k k k 3
.1384389944122896311... 2 212
422
0
1
log ( ) log
K k
k
dk
k GR 8.145
... 9 3 9 3
4
13
231
log ( )!
( )!
k
k kk
... log 2
5
.13867292839014782042... 4 2 2 2 31
16
2
1
log log( )
kk k
k
k
=( )
( )!
2 1
2 41
k
k kkk
!!
... 2 21 12 1
4 1
2
2 21
Gk
k k
k
k
log( ) ( )
( )
.1388400918174489452...
16 2 1640
dx
x
.138888888888888888888 = 5
36
1
2 3
1
51
1
1
k k k
k
k
k
kk( )( )
( )
= k
k k k kk
2
1 1 2 3( )( )( )( 4)
2 .13912111551783417702... e pf k
kk
( )
20
... 14
5
1
2
1
2 1
1
1
arcsinh( ) !
( )
k !
!k
k k
k
... 2 1 2 1 2 2 2arcsinh log log 2
=( ) ( )!!
( )! ( )
1 2 1
2 2 1
1
1
k
k
k
k k k
.13920767329150225896... 1
31
2
4 2 16
2 2
20
4
log tan
cos
/
Gx x
xdx
2
GR 3.839.4
... 1
6 2 6
3
2
1
2 3
1
21
csch( )k
k k
.13940279264033098825... 2
1 12
1
erf e dxx( )
... e
e x
dx
x x
dx
x4
5
41
1 1
2
12
17
14
sinh sinh
... )3(2
5
2 .13946638410409682249...
4
1,
2
3,2,2
3
4
39
412 F
=
11
2
12)!12(
)!(
kk
k
kk
k
k
... sec ( )( )!
1
21
2 40
2
k
k
kk
E
k
... log log2
3
3
12 42 2
2
H kk
k
3 .13972346501305816133... log e e
.1398233212546408378... 1
5
1 2 1
5 2 20
1
cos sin sin
e
x dx
e x
=7
50
=1
2 71
0
2 1 7
cosh log( ) ( ) log
k
k
ke J943
... 1
4
... 1
121 e k
k
Berndt 6.14.1
... 5
62
1
2 1 2
1
41 0
log( )
( )
k k kk
k
k
= ( )( )
11
22
kk
k
k
=dx
x x5 41
... 31
3
1
3 2 10
arctanh
kk k( )
=9
64 91
kk
k
... ( )9 ... e i i 2 Shown transcendental by Gelfond in 1934
...
2 2
2
1
sinh( )
k
k
k
k !
... e
.14111423493159236387... 1
2 221
kk
k
.14114512672306050551...
1
8
1
1 1 1 13 3 3 3
3
1
logcos
e e e e
k
ki i i ik
...
Lik k
k
kk
k
k2
22
2
2 2 2 1
( )
7 .1414284285428499980... 51
... 2
32 2 4 3 4
1
2
22
2
2
12
30
1
log loglog
( )Li
x
xdx
... 3 2 32 3 1
1
1log log ( ) /
k k
k
... 323
2
77
6
1
2 50
log( )
( )
k
kk k
.1415926535798932384...
3 16
1
( )sink
k
k
k
1 .1415926535798932384...
2 14
1
( )sink
k
k
k
= arccos2
0
1
x dx
= x x d2
0
2
sin/
x
3 .1415926535798932384...
= 1
2
1
2,
=3 1
41
8
4 1 4 31 1
k
kk k
kk k
( )( )( )
Vardi, p. 158
=dx
x
e dx
e
x
x1 12
2
/
=x
xdx
10 sin
=dx
x2 20 cos
= log log1 13
12
02
0
x dxx
dx
= log( )1 22
0
xdx
x GR 4.295.3
=
02
22
1
)1(logdx
x
ex
= log22
0
121
1
x
xx dx GR 4.298.20
=
02
11loglog dx
xx
=
02
22
1
))1(log(dx
x
ex
= 1
0
2logarccsc dxxx
.1415926535798932384...
3 2 23
2
1
2
222 1
1
Fkk
k
k
k
, , ,
= 122214
15625 6
6
1
kk
k
.141749006226296033507... log ( ) log2 2
0
1
1 x x dx
... log log2
2 32 4
2
8
2
=
40
4
x x x dtan tan
/
x GR 3.797.1
...
e
B
kk
k
k
1 0 ! J152
.14189705460416392281... arctan( )
( )
1
7
1
7 2 12 10
k
kk k
=
1
21
)2(
2arctan)1(
k
k
k
[Ramanujan] Berndt Ch. 2, Eq. 7.5
= arctan( )
1
2 3 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
... 7 3
2 28
1 1
2
2
12
( ) ( )( ( ) )
k k kk
k
.142023809668304780503... ( )( )
17 1
1
1
kk
k
k
... 4
1
43
4
2 22 1
2 41
log
( )!!
( )! ( )
k
k k kk 1
... 4
11
.142514197935710915709... 6 42
4
2
46
1
2
21 3 2
4
2 3 4
4
( )log log ( ) log
Li
= log ( ) log3
0
1 3
1
21
1
x
xdx
x
xdx
.14260686462899218614...
1 2
)()(
kk
kk
1 .14267405371701917447... 2
22 2
125 5
1
5 1
1
5 4csc
( ) ( )
k kk
... 8
1
4 12 21
dx
x( )
.1428050537203828853... 14
81
2
27
2
31
3
1
log ( )k kk
k
H k
2
... ( ) ( )
1 1 1 12
22
2
k
k k
k
k kLi
k
=1
7
...
3G
... 3
216
... ( )
logk
k k kk k
1
1
1 11
1
32
2 k
... ( )k
Fkk
2
.14384103622589046372... 1
2
4
3
1
7
1
7 2 12 10
log( )
arctanhk
k k K146
= dx
x x
dx
x x x32 0 1 2
( )( )( 3)
... 5
2
4
14 2
e
x
xdx
cos
( )2
... e e ek
ke e
k
i icos cos(sin )cos
!1
0
11
2
GR 1.449.1
= )1sinh(cos)1cosh(cos)1cos(sin
.14430391622538321515... 4
2
0
x e x dxx log
3 .1434583548180913141... log e e
1 .14407812827660760234... Li2
5
6
1 .14411512680284093362...
1
2
2
)14(
18
4222log3
k kk
kG
= ( ) ( )k k
kk
1 1
41
.14417037552999332259... Likk
k4 4
1
1
7
1
7
... 23
2
2
3
1
2 1
1
1
log( )
( )
k
kk
k
k
.14430391622538321515... 4
4 2
0
e edx
xx x GR 3.469.3
=
e x x dxx 2
0
log
3 .14439095907643773669... 2 2
2
1( )kk
.14448481337205342354... e e ek
k ee e
k
kk
1 1
1
1
11
1/ / ( )
( )!
.14449574963534399955... H
kk
kk
( )
( )
3
1 3 2 1
2 .14457270072683366581... 2
41
k
k k k!
.144729885849400174143...
12 )2(4
)2(loglog1
kk kk
k
e
...
0
2/1log
2
1)1(log
kk
k
k
k
k Prud. 5.5.1.17
... 31 5 32 53
2
11
25
1
( ) ,( )
kk
... 1 1 1 2
2 221
1
1k k
k
kk
k
k
cos
( ) ( )
( )!
...
2
3 22
1
16
3
2
11 2
k k
kk
k
k
( ) ( ) 1
=H
k k
k kk
kk
k2
2 1
2 2
4
2
( ) ( )
=4
1 12 2 20
x
x edxx( ) ( )
= log( )1 2
0
1
xdx
x GR 4.29.1
...
2
26
1
21 1
( ) ( ) ( )k
k
k k 2
... 2 3/
... 381
2187
1
2
1 7
1
( )k
kk
k
... 2 112
21
log( )eke k
k
GR 1.513.3
.14544838683609430704... 1
25
3
58 25
8 5
5
1
25
7
51 2
21
( ) ( )
=1
5 2 21 ( )kk
... Likk
k3 3
1
1
7
1
73 0
1
7
, ,
... 2 3 2
12 32 18
3
2
1
3
3
162
2
3 4
7 3
4
GGlog
( )
= H
kk
k ( )4 3 21
...
0
2coslog442arctan85log3100
1
25
11dxxxex x
... ( )
!( )!
k
k kk I
k kk k
1
21
1
221
1
=7
48
1
4
1
1 5
1
421 0
1
23
k k k k kk k
k
k( )( )
( )
= x x d3
0
1
1log( ) x
= log 11
15
x
dx
x
... )2(56)4(140)6(56log24
53
=
1 4
1)2(
k k
k
... log ( )!!
( )!!
2
2
1
4
2 1
2
2 2
2
1
2 22
2
G k
k kk J385
... 4
... 2 22
21
1
2
2 1
1
loglog ( )
k
k
k
H
k
.1463045386077697265... 44
2 21
2 1
2
21
log
( )k kk
GR 0.236.7
= log( ) log1 2
0
1
x x dx
... 2 21
2
3
22
1
2 1 2
1
1
arctan log( )
( )
k
kk k k
... 1
842 kk
... cos
/
x
edxx 2
0 1
... 4
22
22
1
2
1
... 1
2
2 1 1
2 21
cos sin log sin log
e
x x
xdx
e
= xe x dxx sin0
1
1 .14649907252864280790... 1
21 k kk !
2 .146592370069285194862... 6
52
3 20
2
cos
/ cos
x
xdx
... 24 13 1 20 11
2 2
1
1
cos sin( )
( )!( )
k
k
k
k k
... 1
121 ek
k/
Berndt 6.14.1
.14711677137965943279... )2()2(4
)1()1( iii
=
222
1
1
11)12()1(
kk
k
k
kkk
=
0 )1(
sin
2
1dx
ee
xxxx
.1471503722317380396... ( )
1
2 2
1
1
k
kk
...
0
6
3)cos(sin
10
1
40
3log9dx
x
xxx Prud. 5.2.29.24
.14722067695924125830...
loglog log
2 2
2
2
2 122 3
1
2
Li
= log log log2 3 21
2
1
42
2
Li
= log( )1
20
1
x
xdx
... 90
90
1
4 14
( )
.14726215563702155805... 3
643
0
1 x xarccos dx
... 11 1
111
Ei
e
H
kk k
k
( )( )
( )!
= dxxxex 1
0
1 log
.14747016658015036982... ( ) ( )( )
( )
1 3 11
11 2
1
3 3
3 32
k
k k
k kk k
k
... 1
9 1
3
931 1k
k
kk
k
( )
... 58 1
2 30
e k
k
kk
( )
( )!
.14779357469631903702... 2
2
1
21
arcsinh
=( ) ( )!!
( )! ( )
1 2 1
2 4 1
1
21
k
k
k
k k
1 .14779357469631903702... 2
2
1
21 2
0
1
arcsinh
x dx
.147822241805417... ( ) log
1
122
k
k
k
k
.14791843300216453709... 3
23 1
1
9 31
log
k kk
J375
= ( )2 1
91
kk
k
... 4 31 4
1 4
0
ek
k kk
/
( )!
1 .14795487733873058... H5 23
/( )
.14797291388012184113... 2
42
1
4
1
22 21
coth
kk
1 .14812009937000665077... 1
7
2
8 2
1
4 421
tan
k kk 7
... ( )k
kkk 21
=4
27
1 1
20
( ) ( )k
kk
k
... ( )
( !)
k
k kI
kkk
2 12
12 2 0
10
... Likk
k2 2
1
1
7
1
72 0
1
7
, ,
... 4 21
2
10
3
1
2 2 50
arctan( )
kk k
... 3
27
... 13
18
3
2
1
3
1
3
1
3
1
3 21
loghg
k kk
=( ) ( )
1
32
k
kk
k
=dx
x x x4 31
2
... 1
4
3
4
1 2
220
e k k
k k
k
( )
!( )
... 21 5/
... 16 7 4 31
16 1
2 2
0
( )( )
kk k
k
k
... 41
22
23
2
3 21
Lix
x xdx
log
.14900914406163310702... 21
2
1
3
1 2 1
2 3
1
1
log( ) ( )!!
( )!
k
kk
k
k k
... logk
k kk3 2
2
... 4 7
9
3
2
3 2
1
e k
k
k
k
( )!
... 93
2
7
2
1
2 1 20
log( )
( )( )(
k
kk k k k 3)
=
0 )2(3
1
2
1
kk k
... 8
19 42 2
0
2
ee e xx
/ sin cos
x dx
...
1
13 !
3)1(
3
k
kk
k
k
e
6 .14968813538870263774... 2 2 33
2
1
4
1
2 12 11
Fk k
kk
, , ,!( )!
( )
!
... 19
5 23 2
1
4 1
1 1
421
2
log
( )
( ) ( ( ) )
k k
k
k
k
kk
1 .14997961547450537807... 2 13
2
k
k
k ( )
.15000000000000000000 = 20
3
.150076728375356585727...
22 2
11log
2
1
)1(2
1)()1(
kkk
k
kkk
k
.15017325550213874759... 2
3
3
21
3
4 21
sin
kk
...
1
2
1
13 )1(
)1(
)2(
1
8
)3(
k
kk
k k
H
k
= dxx
xxdx
x
x
1
0
1
0
22
1
log)1log()1(log
= 1
0
1
0 1
)1log(dydx
xy
xy
... 24
)21log(228
5
8
1
16
1 2)1()1(
Berndt 3.3.2
= 12(21(2 22 iLiiLii
= 2
2 2 1
2
20
k
k
k
k k
( !)
( )!( )
= 22 2
0
2 x x
xdx
sin
cos
/
... 1
21 1
1
2 11
sin cos( )
( )
k
k
k
k !
= sin1
2 51 x
dx
x
... 3 3
4 15
2
42 2
2
4
4 2 23
4 ( ) loglog
log
6
1
2
21
43 24Li ( ) log
=log ( )
( )
3
20
1 1
1
x
x xdx
... 8 2 4 3 11
3 11
log log( )
( )
k
kk k k
... 1
4 2 6
2
3
1
3 2
1
21
csch
( )k
k k
... 1)1(1)(2
kkk
...
03 3
)1(
k
k
k
...
1
2
2 1010
10,2,
10
1
729
110
kk
kLi
...
12
2
)4)(2(36
113
3 k kk
k
1 .15098236809467638636... 3
10
3 3
10
2 2
5
1
6 5
1
1 6 121 0
log log
( )(k k k k )k k
= dxx
x
1
06
6 )1log(
.15114994701951815422...
12 3 32 20
dx
x( )
...
127k
kk
k
H
.1511911868771830003...
( )4 1
2 14 11
kk
k
81 .15123429216781268058... 8
.15128172040710543908...
1
01
3
)()1(
kk
k k
1 .15129254649702284201... 10log2
1
.15129773663885396750...
13
1
k kkF
.151632664928158355901...
1
21
2!)1(
4
1
kk
k
k
k
e
1 .15163948007840140449... ( ) ( )
( )
3 5
4
2 .15168401595131957998...
1 22/52/5 )1(
1
k k k
k
k
k
.151696880108669610301... 25
4
2csch
162coth
82
2
=
222
2
1
1
)14(
4
4
1)2()1(
kkk
k
k
kk
.15169744087717637782...
82 2 1 2
0
2
( log ) log(sin ) sin/
x x dx GR 4.384.9
.151822325947027200439... 1
2log
e
.15189472903279400784...
1
2
1
32
)1(
3
2log
69
4
k
k
kk
=4
12
11log
x
dx
x
1 .151912873455946838843... 1
16 5 4 3 20 k k k k k kk
.1519346306616288552... 5
6
6
51
51
1
log ( )
k kk
k
H
... log ( ) k
kk2
2
... 2 5 21
22 1
1
1
log1+ 5
2
( )
( )
k
k k
kk k
...
1
2)12(
61
2
3cos
5
1
k k
...
1
22
)12)(1(3
2log2
3
2log4
18 k
k
kk
H
.1523809523809523809 =)4)(3(4
12
105
16
0
kkk
kk
k
...
1 87
8log
7
8
kkkH
...
0
2
)7)(4(!
2
42
1
k
k
kkk
e
... 2)1(
1 .15281481352532739811...
1 3!)!12(
!)!2(
3
1arcsin
4
23
2
1
kkk
k
...
21
1
kkk
k
.15289756126777578495...
1
2
1
2
3
5
)1(
)1(dx
e
x
e
k
ee
eex
kk
...
13
234 )1(
11)3()2(3
kk kkkk
=
1
1)3(k
k
...
2
2)2()()3()2(4k
kk
...
3
1
3
1
2
1
3cot
322
1
=
113 3
)12()2(
3
1
kk
k
kk
kk
k
... )3(
)3(log
... 2 2
10
1
six
xdx
arcsin
arccos
... )1(Im2)1()1(1coth iiii
=
1
11
k ikik
...
12 1
21coth
k k J947
... coth ( ) ( ) Im ( ) 1 2i i i i
... 1 2
2
1
2 1 2 2 1
1
1
log ( )
( ) (
k
)k k k k GR 0.238.2, J237
= kk
kk
log2 1
2 11
1
J128
=
1 )48(
1
k kk
= 1
2 22
kk k k( )
= ( )
( )!
2 3
22
k k
k
!
k
= ( )
1
2 40
k
k k K Ex. 109b
= ( )
1
8 2
1
31
k
k k k
=
12
2
1 4
11log41
)12(4
)2(
kkk k
kk
k
= dx
x xx dx5 3
1
3
0
4
tan/
=
02
3
sinlogsin1
sindxx
x
x Prud. 2.6.34.44
= log
logx
xdx
x
x
dx
x21
2
2 1
=
1
02
12 )1(
)1log(
)1(
11log dx
x
x
x
dx
x GR 4.291.14
= ( ) arctan10
1
x x dx
= x
dx
xx log2
1
log2
1
1
11
02
GR 4.283.5
=
1
2
1 2
0 xe
e
x
dx
xx
x /
GR 3.438.1
= dx
e ex x2 20 1( )
1 .15344961551374677440...
0
2
4
0211 )!()1()2()2()2(3)2(2
k
k
k
kJJJJ
=
1
1 2
)!12(
)1(
k
k
k
k
k
= 13
2
... k
k kk
2
2 11 1
.15354517795933754758... )7()1(
.1535170865068480981...
1
1
13 7
)3()1(
17
1
kk
k
k
k
k
... 7/1e ... 8/1
... )1(360)2(6)3(42
2log
590
181
=
2 4
1)(
k k
k
.15409235403694969975...
1
22
2
1
2
)112(
12
12
)2(
32csc
3sin3
48 kkk k
kkk
.1541138063191885708...
23
3
1
)1(
2)1)()(2)(1()1(
4
9)3(2
kk
k
kkkk
= 3)2(
=
134
2log
xx
x
= x x
xdx
2
0
1
1
log
= x
e edxx x
2
20 1
.154142969550249868...
1
)3(
)1(4kk
k
k
H
.15415067982725830429...
11 7
1
7
16log7log
kk k
Li
=
0
12 )12(13
12
13
1arctanh2
kk k
K148
.15421256876702122842... 2
2 2164
1
2 1 4
(( ) )kk
J383
15 .15426224147926418976...
0 !k
ke
k
ee Not known to be transcendental
=
0
!/1
k
ke
... 2 12
1 2 11 2 1
12 2
k
kk
k
k
k kk k
( ) log( )
... 2
... 3 10
... ( ) log ( )
1 21
1
k
k
k 1
... 2/3)2(
2
39
40H
... ( )
log
13
2
k
k k k
...
1 2 ),2(
1
k kkS
... 7
32 2
1
4 2 1
23
1
( )
( )
k
kk
k
k
k
k
.15468248282360638137...
13 1)13(
1
k k
=
6
35332
6
3534
3318
1
6
3log
3189
ii
i
i
.1547005383792515290... 2
31
1
2 1
21
0
( )
( )
k
kk
k
k
k
k
=( )
( )!
2 1
2 41
k
k kk
!!
...
0
2
16
1
3csc
3
2
kk k
k AS 4.3.46, CFG B4
=
122!)!2(
!)!12(1
kkk
k J166
.15476190476190476190 =
0 )103)(13(
1
84
13
k kk K134
... 3 61 3
0
1
e ex /
dx
...
1
4
)!1(13
k k
ke
...
1 !
)1(3
k k
kke
.15491933384829667541... 1
5
3
51
2
61
1
( )k
kk
k
k
k
... )(Re i
...
3
2
...
122
2 11 2
k kkH
... )3()2(
)3()2(
... )3(2log723log54318216 2
=
1
1
134 6
)3()1(
6
1
kk
k
k
k
kk
... dtt
tdx
x
x
e
0222 1
cos2
)1(
cos
...
2 202
121
1
)!(
1)()1(
k k
k
kJ
kk
k
.15580498523890464859... 4 4 2 22
3
1
22
arctan
1
2log Li
=( )
( )
1
2 2 1
1
21
k
kk k k
.1559436947653744735... cos ( )( )
2 12
20
!
kk
k k GR 1.411.3
.15609765541856722578... dx
x x4 31
x
...
kk
kk
kk
k
kk
11log2
11log221
4
1
1
1)( 2
122
.15622977483540679645...
1 2
)(
kk
k kH
.156344287479823877804...
7
10 4 21
2 1
41
1
coth ( )
( )kk
k
k
=
222 2
1)(Re14
1
k
k
k
ik
k
.15641068822825414085...
dx
x
xdx
x
x
e 1
3cos
)1(
3cos2
0223
.15651764274966565182...
1 0
1
221
22 !
2
1
11
2
11coth
1
1
k k
kk
kk k
B
kee
1 .15651764274966565182...
022
022
2
1
11
2
1coth1
1 kkk kee
e
J951
...
1
1
)2()!2(
)1(
2
231sin101cos6
k
k
kk
1 .15654977606425149905...
1
22cos
!
sin
42)2cos(sin
2
122
k
ee
k
keeeee
ii
...
3 3
0
6 x x dsin x
... 2
11
2 2 20
log
log log
ex
e xdx
... 3
6
3
41
13
13
logx
dx
x
...
05520 xx ee
dx
... 1
33 e
3 .157465672184835229312...
0
2
2
1
)3log()31log( dx
x
x
1 .15753627711664634890...
13
133 )13(
1
)13(
11
27
)3(26
kk kkG
1 .15757868669705850021...
4
3
2 1
2
0
2
x dx
ex
... 1
9
2
6
1
3 1
21
21
( ) logx x
x xdx GR 4.233.3
... cos ( )( )!
3
0
2
11
41
3 3
2
k
k
k
k GR 1.412.4
.15779670004249836201...
2 21
kk
k
...
1
21
/12 !)1(
1
kk
k
k
k
e
... log8
.157894736842105263 = 19
3
.15803013970713941960... 1
4
1
4
14 9
1
e x
dx
xcosh
.1580762... 1
1 2
1( )
int
a non trivialegerpower
...
12
21
02
2
)1(
log
)1(
log)2(
2
)3(3
xx
xdx
x
xx
...
1 )12(3kk
k
k
H
...
0 1 ),2(
!
k kkS
k
...
1
2
1
14
)1(
2csch
42
1
k
k
k
.158747447059348972898...
log ( )( )
22
22
12 1
41
1
i i k
kk
kk
...
1
2
1
3
)1(3csch
6
3
6
1
k
k
k
J125
...
14
12
2
1
log
)14(
1
216 x
dxx
k
G
k
...
1 )2)(1(242log6
kk kk
k
... 6 21
13
230
log( )
( )( )
k
k k k
...
1
2
22log68
kk
kHk
40 .15890107530112852685...
1
2
k kF
k
1 .1589416532550922853... B
k
k
kk
k
2
0 2
2
( )!
3 .15898367510013644053...
1 1 1 111 1 11 1
0
1
4
2
1
12
1
12
)(
2
)(
i j k liijkl
j kijk
i jkj
kk
kkt
... 3;1;)!(
3)32( 1
020 F
kI o
k
k
.15909335280743421091... 1
0
6 arcsin245
16
14dxxx
GR 4.523.1
.15911902250201529054... 5
2
1
21
1
2 2 1 2 1
1
2 11
arctan( )
( )(
)
k
kk k k
...
1 )2)(12(
1
12
2log1613
k kkk
.1591484528128319195... ( ) log arctan
18
1
94
9
82 2
1
2 21 2
1
k kk
k
H
... 21
... )5(
)3(
...
1
2
2
)22()2(
kk
kk
.159436126875267470801... 2 2
132
5
12
2 2
3
2
3
1
3
log log ( )
( )
kk
k
H
k k
1
.15956281008284... 7
96
1
2
1
161
2
24
24 1
4
21
Li e Li ek
ki i k
k
( ) ( ) ( )cos
=
1 )4)(1(
1
144
23
k kkk
...
0 )12(
1
4
1cothlog
2arccoth
kk ke
eee
.159868903742430971757... log log ( )
log ( )2
1
22 1
22
2 1
k
kk
k
k
k
k
k
... ( )1 5
6
=
1
1
4
)1(
25
4
kk
k k
.1600718432431485233... 1
54 5 1
41
( log log ) ( )( )
kk
k
k
.16019301354439554287...
112
1
2 76
)1(
6
1
kk
k
kk
k
k
H
kLi
.16043435647123872247...
1 )3(2
1
3
2log7
9
16
kk kk
... dxx
x
e
023 3
cos
32
... 24 2 7
601
14
0
1loglog
x
xdx
.1606027941427883920... 25 1
30
2
0
1 2
21
e k kx e dx
x
xdx
k
k
x( )
!( )
log
=( )
( )! !
1
1 20
k
k k k
...
1 )12)(12(
2cos
2
1
4
1sin
k kk
k GR 1.444.7
... 22
22log
2
242log
2
2
=
12
1
4
)1(
k
k
kk
=2
14
1
0
4 11log)1log(
x
dx
xdxx
... 2
22
2
2
2 2
2 21
14
0
1
log log log
xdx
...
11
1
2
log
2
2log
2
1
2
1)1(
kk
kk
k kLi
k
.1613630697302135426... 1
2
1
21
1 2
22 2 121
2
1
coth( )!
k
B
kk
kk
k
.16137647245029392651...
222
2
1
)!2()!2(
1)2(00
1 kJ
kI
kk
k
k
.161439361571195633610...
1
2122
)!2(
2
2
1log1sinhlog
k
kk
kk
B
e
e
= 2
2
2 12
1
kk
k
B
k k
( )!
=
12
1 )2()1(11log
kk
k
k
kii
1 .161439361571195633610...
1
2
!
2)1(
2
1log1sinhlog1
k
kkk
kk
Be Berndt 5.8.5
... 3 2
20
1
192 1
arctan x dx
x
2 .16196647795827451054...
1 2k
k
k
kk
1 .162037037037037037037 = 3)3(
216
251H
.16208817230500686559...
1
1
3)!2(
)1(1
3
1cos
kk
k
k
... 3 3
2
11
65
60
( )( )k kk
.162162162162162162162 =
0
6log)12()1(6logcosh2
1
37
6
k
kk e J943
= dxe
xx
0
6sin
...
112
2
2
)2(
)12(
122log4
2 kk
k
kk
kk
3 .162277660168379331999... 10
1 .16231908520772565705...
1 21 )!(
11,0,
2
322)1(
k k
eerfe
.162336230032411940316...
1
3
)!3(2
7113
k k
ke
...
15
21
4
)(6
)5(93
k k
k
... 2/12
12
3
1 22212
1
!
)()1( kekk
kk
k k
kk
k
...
1
3 )12(2)12(27
)1(
812
k
k
kk
= dxx
xx
1
022 )1(
arcsin GR 4.512.7
.16268207245178092744... 2 2 31
8 4 31
2
log log dx
x x
.16277007036916213982... 2 31
22
4
5
1
2 2 1 2 1
1
2 11
arctan log( )
( ) ( )
k
kk k k k
... 3
1
.162865005917789330363...
1
0222
22
1
)1log(
2
1
2
1
8
2log
96
11dx
x
xxiLi
iLi
...
3
2 2 2 23 3
256
1
1024
7
8
5
8
3
8
1
8
( ) ( ) ( ) ( ) ( )
= ( )( ) ( ) ( ) ( )
11
4 1
1
4 2
1
4 3
1
4 43 3 30
k
k k k k k 3
...
4
2
0ex e x x dxx
sin cos
...
( )log ( )
k
kk
k
1
2 1
.1630566660772681888... 7 3 12 1
2
1
e Eik
k kk
( )( )!( )
... ( )
( )
k
kk
1 2
2
...
1 849
8
kk
k
... 2/
0
322
cos27
40
6
dxxx
...
13)3(
)3(3)2(8
17
k k
k
...
1
1
)!3(
)1(
32
3cos
9
2
9
1
k
k
k
ke
e
.163900632837673937287...
1 2
5
2
3log
4
1)2(
8
3log
kk k
k
.16395341373865284877... 3
1
12 2 1
41
e
e kk
2 .16395341373865284877...
0
2
)!2(1
1
2
1coth
k
k
k
B
e
e
4 .16395341373865284877... 3 1
1
12 2 1
40
e
e kk
.16402319032763527735... 1
12
2
3
4
3
3
4 9
1
6
1
31 1
21
( ) ( ) ( )
=k k k
kkk k
( )
( )
2
9
9
9 112 2
1
2 .16408863181124968788...
22 3
)(
k k
k
.1642135623730950488... 25
4
2 1
2 22
( )
( )!
k
k kk
!!
1 .16422971372530337364...
5
4
... loglog
224 4
2
2
1
2
1
2
2 2
2 2
Li
iLi
i
= log( )
( )
1
1
2
20
1
x
x xdx
1 .16429638328046168107...
13)!!(
1
k k
.164351151735278946663...
1
250 43
)1(
3
2log
331
xx
dx
kk
k
.164401953893165429653...
1
2
)1(2
)1(
2
3log
kk
kk
k
H
6 .16441400296897645025... 38
.16447904046849545588...
02
0
1
)2(!
)1(
)1()!1(
)1(1)1(
1
k
k
k
k
kkkk
kEi
e
= dxe
xxx
1
0
log
.16448105293002501181...
k
k
kk
H
kLi
2)1(2
1
2
122log
6
)2(
022
22
... 2
5160
11
logx
dx
x
...
1
0
24 log)1log()4(2
45 x
xx
.164707267714841884974...
13
42
)12(1288
2log)2log1(
8
)3(7
k
k
k
kH
.16482268215827724019...
primepk p
p
k
k
1
loglog
)3(
1
)3(
)3(3
23
... 1
4
.165122266041052985705... ( )( )
16 1
1
1
kk
k
k
... )5()3( 9 .1651513899116800132... 84 2 21
2 .16522134231822621051...
222
1)()1(
1
kk
k
kFkkk
k
= 2
5tan55
102
15
2
5
2
3
...
1
1
)!3(
)1(
2
3cos
3
2
3
11
k
k
ke
e
2 .1653822153269363594... e EiH
kk
k
( )!
11
... )1(
...
23
123
log
)!1(
1
log
1)3(1
k
n
nk k
k
nkk
k
... 2 3 4 2 3 4 24 1
2 2
1
log log log log( )
H
kk
kk
...
0 36
)1(
16
)1(32log
6
3
4 k
kk
kk
= dx
x x2 21 1
.165672052444307559...
0
logcossin5log2arctan210
1dxxxxe x
... 12
12
2
12
1
2
1
2
2 3
3
log logLi
=log2
3 21 2
x dx
x x
.165873416657810296642... e
e
8
)1cos1(sin1sin1cos1sinh1cos1sin1cosh
4
1 2
=
1
1
)!4(
4)1(
k
kk
k
k
.16590906347585196071...
123
2
53
1
250
63
50
3
3050
3log9
k kk
...
1
8
1
42
2
3 2 1 21
( )/ /
log x dx
x x
.16599938905440206094...
14
2
)2(1)4(4)3(4)2(
k k
k
...
1
2
)12(
1
!)!2(
!)!12(1
4
k kk
kG
J385
...
1
1
)!4(
4)1(1cosh1cos1
k
kk
k GR 1.413.2
... log3 2
2
... ( )
( )log2
3
23 2
1
4 2
2
4 31
k
k kk
= ( ) ( )k k
kk
2
22
.16658896190385933716... 2
2
.16662631182952538859...
4/
02cos)cos(sin
sin
8
2log
2
2log
4
dx
xxx
xx GR 3.811.5
.166666666666666666666 =1
6 1
11
4
32 2
12
3
k
k kk k k( )! k
=
1 4
1)2(
kk
k
=
123 )4)(3)(2(/1
1
kk kkk
k
kkk
=
0 )52)(12(
)1(
k
k
kk
=
2 1
1 2
)2(2
)1(
)2(!
1
k kk
k
k
k
kkk
=
32
41
k k
= k k
k k
k k
k kk k
( )
( )( )
( )
( )( )
4
2 2
6
5 11 1
J1061
= arccosh x
xdx
( )1 31
1 .166666666666666666666 =
1
4
)(2 2
)8(
)4(
6
7
k
k
k
HW Thm. 301
=
primep p
p4
4
1
1 Berndt Ch. 5
.16686510441795247511... sinh sin sin
( )
1 1
2 4
1
4
1
2
1
4 30 !
e
e kk
...
2 20
11
2
!!
1)(
k k kkI
kk
k
...
0
4/14/1
)!4(
4)1(cosh)1(cos2cosh2cos
2
1
k
k
k
...
0
2
!
)1(22
k
k
k
ke GR 1.212
... 8
arccos
2 .16736062588226195190... 22
2cos2cosh
=
142
4/34/1 11
)1(sin)1(sin
k k
.167407484127008886452...
1
2
1
6
)1(
13
13log32log
k
k
kk
...
1 )!1(2
11
k k
5 .16771278004997002925... 6
3, volume of the unit sphere in R6
...
loglog ( )
( )
22 1
22
kk
k
k
k Dingle 3.37
= 1
2
2 1
21 k
k
kk
log
.16791444558408471119... 9
2
3
2
1
3
12 2 1
91
cotkk
...
0 )!3(
1
2
3cos
2
3
1
k kee J803
...
1
3
)!3(
))!1((
k k
k
.16809262741929253132... ( )
( )
3 1
3
1 .168156588029466646... log
( )
2
2 2 1
k
k kk
...
1
4 1)3(k
k
...
0
12 )12(6
1
6
1arctanh
kk k
AS 4.5.64, J941
...
6
2
32
5 3
4
3 3 3
1
6
5 3
4
3 3 3
log
i
i
i
i
=1
2 1
3
831 1( )
( )
k
k
kk
k
...
1
1
23
1)13(2)1(2
2
k
kk
k
kk
k
...
144 6
1
6
1
kk k
Li
... dxx
xx
1
02
2
1
1log
322log2
2
3log3
=
12
2
1
1log dx
xx
x
.168547888329363395939... 4 16
1
8 2 1 2 2 1
2
4 21
( ) ( )k kk
.168655097090202230721... 4 2 2 14
2 2 1 8 27 3
16
2
log log( )
=1
8 2 1 2 2 14 31 ( ) ( )k kk
.16870806500348277326...
1
21
1
2 2
1
21
4 2
4
2 1
81
( )kk
k
= 1
8 31 k kk
...
2
1)(
kkk
k
...
5/4
1
5/1
1
...
23 3
)1(
k
k
k
.16945266811965260313... 14
27 92
0
1
x x xarcsin arccos dx
...
32
3cos
2
9
1 e
e
.16948029848741747267... ( )k
kk
4
42
= 1
1
1 14
24 4
212 4
220 4
2k k k k k k
1
kk k k k
= ( ) ( ) ( ) ( )4 1 8 4 1 16 4 1 16 4 11 1 1 1
k k k kk k k k
.16951227828754808073... 21
2
12 2 1
41
cotkk
=
112
1tan
2
1
kkk
Berndt ch. 31
... 9log2
11 .1699273161019477314... 12 22 6 2 23 2
0
x xd
ex
log x
...
0 )12(6
12
2
3arcsin
2
3
kk kk
k
... k
k
kk
1log
1
22
...
133 6
1
6
1
kk k
Li
...
1
0
2
2
1
1)1log(
121
xe
dxx
eLie
...
2
1)(k
k
... 4/34/34/5
2coth2cot214
23
=
12
41)4(8
8
8
k
k
k
kk
...
k
k
k
k
k 9
)2()1(
19
1
2
1
3coth
61
12
...
12
)1(
)1(3
16
3
2
9
1
12
5
362
3log
k kk
k
=
1 3
)1)((
kk
kk
... kkF
8
3
.1710822479183635679... ( )k
kk 41
...
1)(
1)13(
8
1
k
k
... Re ( ) ( ) i i i
...
0 )!2(3
1
3
1cosh
kk k
.17138335479471051464... 41 1)1()1( kk
...
12
1 1( ) ( )k kk
1 .1714235822309350626...
1
428
)()4(8100 k
kkd Titchmarsh 1.2.1
...
1
2
5
41
21
log coscosk
kkk
.1715728752538099024... 3 2 22 1
8 1
2 1
2 2 11 1
k
k k
k
k kkk
kk( )
( )!!
( )! ( )
=( ) ( )!!
( )! ( )
( )
( )
1 2 1
2 1
2 1
4 1
1 1
11
k k
kkk
k
k k
k
k k
=
0
)2()!2(
)1(
2
1
k
k
kk
... ( )
( )
2 1
1
112
1
22 2
2
k
kk Li
kk k
.17172241473708489791... 7
)3(
... 6 3 1 5 13
1
cos sinlog coslogx x
xdx
e
.17186598552400983788... 11
22 2( ) ( )i i
= 1)1()1(2
11)()(
2
1 iiii
= 11 2 13
2
1
1k kk
k
k
k
( ) ( ) 1
=
1
)14()14(k
kk
= 1
10
cosx
e edxx x
.17190806304093687756... dx
x x x4 21
...
i x
e ex x
3
42 1 1
12 1 3 2 2 3
0
( ) / ( ) /( ) ( )dx
GR 3.418.1
= 1
3
1
3
2
3 11
2
20
1
( ) log
x
x xdx GR 4.23.2
5 .172029132381426797... dxx
x
1
03
32
)1(
log
4
)3(9
1 .1720419066693079021...
1
3)1(4
11
2sinh
5
8
k k
...
22 2
log
k k
k
...
1
2/1)1(4
tanh2
1
2
1
k
kk e J944
... 2e
...
1
2
)!3(k k
k
.17229635526084760319... ( ) ( )
( )
( )2 3
4
4 1 31
H
k kk
k
... 1)2(49
4
16 1
2
kk
kk
.17250070367941164573... 11
2
1
2
12 2 1
21
cotkk
... dxxx
li )log3sin(1
23log3
10
1 1
0
GR 6.213.1
.17254456941296968757...
1
1
223
3
1)3()1()1( k
k
k
kkk
k
...
13
1
k kF
...
1
2
)2(32
2
3log4
2
3log3
2
3
kk
k
k
H
.17260374626909167851...
1
21
12 2)!2(
4)1()2(1sinlog
k
kk
k
kk kk
B
k
k
AS 4.3.71, Berndt Ch. 5
... ( ( ))2
...
12
23
11)()1(
kk
k
kkLi
k
k
...
1
1
)3(!
)1(
3
55
k
k
kke
.1728274509745820502... log ( )2
2
1
2 1
1
0
GH
k
kk
k
Adamchik (24)
= dxx
x
1
02
2
1
)1log( GR 4.295.5
...
2 2
log
kk k
k
1 .17303552576131315974...
1 2
cot22
11)2()2(
k k kkkk
...
1
1
)!1(
)1()1(2)1(
2
k
kk
k
kH
e
EiEi
e
...
n
mn
nmhn 1
)(1
lim)3(
)2/3(
h(m) = lowest exponsent in prime factorization of m.
... 11
22 2 e ei i
k
= ( ) coskk
1 11
.17328679513998632735...
1 )14)(24)(34(
1
4
2log
k kkk J253
=
0 44
)1(
k
k
k
=
1 12)14(
1
r srs J1124
= 1
8 2 1 2 2 131 ( ) ( )k kk
[Ramanujan] Berndt Ch. 2
=
15 xx
dx
= dxx
xx
1
022 )1(
log GR 4.234.2
...
12!!
1
k kk
...
2 !
1)(
k kk
k
.17352520779833572367...
03 32
)1(
k
k
k
...
1 !
)1()1(
k k
keEi
... kk)6(
1
...
14)1(
)5(2)3()2(4
)4(
k
k
k
kH
...
1 )13(3
1
kkk k
.17417956274165000788...
1
1
122 5
)1(
6
1
6
1
kk
kk
kk k
H
kLi
.17426680583299550083... arcsinh11
2 1 1
2
2 20
1
x dx
x x( )
...
2
3 216
3 3 9 3 6 4 361
6
log logk kk
= ( )( )
12
61
1
kk
k
k
2 .17454635174140...
1
/11)1(
k
kk
k
e
...
12 2/1
11
k k
...
1
41)(k
k
... )3(log)(1
13
kk
k
p kprimep
.1750000000000000000 = 40
7
1 .1752011936438014569... iiee
sin2
1sinh
2
1cosh2
21sinh
1
=1
2 10 ( )k !k
AS 4.5.62, GR 1.411.2
= 112 2
1
kk
GR 1.431.2
= cosh1
12x
dx
x
.17542341873682316960... cosx
edxx
20
.175497994056966649393...
1 )!1(
log
k k
k
... 1
121 e
k
k
.175639364649... 1
221 n
k
k
n
kkn !
log
!
.17575073121372975946... 83
8 20
1
e
x x
edxx
sinh
.175781250000000000000 = 45
256 9
2
1
kk
k
.17592265828206878168... eFkk
e
k
kk
;3;2
1
)!2(!
)1(10
0
.17593361191311075521...
22 2
11log
2
1
)1(2
1)(
kkk kkk
k
1 .1759460993017423069... i i i x
xdx
4
1
2
1
21 1
0
( ) ( ) sin
sinh
x
.17606065374881092046...
1
1
13 6
)3()1(
16
1
kk
k
k
k
k
.176081477370773117343...
1 3)!2(3
1sinh
32
1
kkk
k
... 4 3
20
3
08 1
log
sinh
x dx
x
x dx
x
1 .17624173838258275887... )(
1 .176270818259917... smallest known Salem number
.176394350735484192865... 20
1
20
2
10 61
log arctan x
xdx
1 .176434685750034331624...
12
1
2
)1(1
k
k
k
.176522118686148379106... 3
2 2
8
21
1 1
21
1
log( )
( )k
k
k
k
= 1
21
11
2 kk
kk
log
.176528539860746230765... 12log2/11
2/11log
24
3
=
0 )3)(2)(1(2
1
kk kkk
J279
.17659040885772532883...
1
1
13
)1(
kk
k
.1767766952966368811... 1
4 2
1
4
2
0
( )k
kk
k k
k
... /1e
.17709857491700906705... 5 1 3 11
2 1 2 31
cos sin( )
( )!(
)
k
k k k
=log sin log
sin3
15
1
1x x
xdx
x
dx
x
e
1 .177286727908419879284... 1
2
2 1)(
k
k
k
.17729989403903630843...
0
1
)53)(13(
)1(
8
1
36 k
k
kk
1 .17743789377685370624...
0102 2)!2(!
12
2
1;3;)2(4
kkkk
FI
.177532966575886781764...
12
22
2
)2(
1)1(
121
kk
k
kkk
=
1
22
22
2
2cos)1(
2
1
k
kii
k
keLieLi
=
02
12
1
0 )1(
loglog)1log(
xx ee
dxxdx
xx
xdxxxx GR 3.411.10
= x
e edxx x
10
=
x x
xdx
log
10
1
=
1
0
1
0 1dydx
xy
yx
.17756041552698560584... 1
82 3 2 3 3
2 11
log( )cos
k
kk
J513
2 .1775860903036021305... 2log
= dxx
x
1
121
)1log(
=
02
2
02
2
4
)4log(
1
)1log(dx
x
xdx
x
x
= dxx
x
02
2
1
)1log( GR 4.295.15
= 2
0
2
1
1log
x
dx
x
x
GR 4.298.9
=
0
tan dxxx
= 2/
02
2
sin
dxx
x GR 3.837.1
=
0
2
1
1
arctanarcsindx
x
xdx
x
x
=
2/
00
)tan4log(2
sinlog
dxxdxx
= 4)3(
1728
2035H
1 .17766403002319739668... 7
...
012 )12(2)!12(
12log
2
1
kk kk
ci
...
1
3
)!3(k k
k
.177860854725449691202...
2
3
3 6
2 3
1( ) ( )
( )
n , where n has an odd number of prime factors
[Ramanujan] Berndt Ch. 5
... 4/
02
22
cos
tan
4162
2log dx
x
xx GR 3.839.3
...
12 1
1
kke
Berndt 6.14.4
... !4log
1 .17809724509617246442... 21arctan8
3
= ( )
( )sin
1
2 1
6 3
2
1
21
k
k k
k J523
= ( )
( )( )
k
k k
k k
k kk k
1 4
4 8
2
12
321
2
21
8 4
3
= dx
x( )2 30 1
= x
xdx
3 2
0
1
1
/
GR 3.226.2
= sin3
0
x
xdx
GR 3.827.4
.1781397802902698674... 12 24
2 2
20
log sinx x
xdx
...
126k
kk
k
H
... ( ) ( )
( )(
2
2 4
1
2
1
1 21
)
k
k k kk
... cosh( )
22
2
2
2 2
0
!
e e
k
k
k
GR 1.411.2
... k
kk
2
2 !!
...
1
2
!!2
2
1
233
k k
kerf
ee
.1782494456033578066... 1
2 42 4 2
1
2
arctan
dx
x x
...
0
4
242
)22(
)12(
4
)3(
14424 k k
k
... k k k
kk
! ! !
( )!31
... 1
162
12 1 1
2
1
log ( )k
kk
k
...
0 )5)(2(!
13
3
25
k kkke
.1785148410513678046... 4
5
5
41
41
1
log ( )
k kk
k
H
.1787967688915270398...
4 3
3
4
1
3 3 1 3 21
log
( )(k k k )k
J250
= dx
x x4 21 1
... 21
21
1
2
20
1
arctantan sin
cos
x
xdx
... 2
5 5
1 21
21
( ) ( )!
( !)
k
k
k k
k
2 .17894802393254433150...
2
21)(2k
k k
.178953692032834648497... 1
2
1
2
1
12 21
cot
kk GKP eq. 6.88
=
1
2121
12 )!2(
2)1()2(
k
kkk
kk k
Bk
... 8
1 11
41
3erfi erfx
dx
x
sinh
... 5 2 61
1 2
1 12
1
( ) ( )!
( )!
k
kk
k
k
...
0 )4(2
)1(
2
3log16
3
20
kk
k
k
.1792683857302641886... 2 3
4
1
2
1
41
2
20
1
log
log( )e e
dx
e ex x
.17928754997141122017...
2 1)(
1)()1(
k
k
k
k
.179374078734017181962... ee
ke
k k
k
11
1
1
1
( )
( )!
.17938430648086754802... 1 1 11
11
( ) log( ) log( )
k
k k k J149
376 .179434614259650062029... 145739620510
387420489 10
10
1
kk
k
.1795075337635633909... 2 2 2 2
204 2 24
2
2
2
2
2
4
log log log logx xd
e x
x
5 .179610631848751409867... e
1 .17963464938133757842... 1
6 4 3
3
2
1
4 321
cot
kk
... )32log(2
3
4
3
3
1arctanh
2
3
4
3
=
0 )12(3kk k
k
... 3
1
35
6
2 22 1
2 31
log
( )!!
( )! ( )
k
k k kk 1
... 7
6
7
6 71
log
Hkk
k
.180000000000000000000 = 50
9
... 5 3
16
2
0
1 ( ) arcsin arccos
x x
xdx
... ( )
! (
1
4 2 1
1
1
k
kk k k )
Titchmarsh 14.32.3
...
0
2
2 32
12
4
3 kkk
k
.1804080208620997292... 23 1
1 20
e
k
k
k
kk
( )
( )!
.18064575946380647538... 50
10 5 51
201 5 2 5 2 ( ) arccsch arcsinh
=
log5 1
32
= ( )
1
5 1
1
1
k
k k
5 .180668317897115748417... )2(e
.18067126259065494279...
1
222
22
)58(
1
)38(
1
8
3csc
642216 k kk
= dxxx
xx
1
042
2
)1)(1(
log GR 4.234.5, Prud. 2.6.8.7
.1809494085014393275...
8 3
3
24 831
log dx
x
... cosh
sinh1 1
3
13
14
x
dx
x
...
2 12
21
kk
...
21 2
11log2
1
)1(2
1)1(
kkk kkk
k
3 .18127370927465812677...
1
1
2
)!12(
1)2(2
k k
k
kI
.18132295573711532536... 6)1(
1 .18136041286564598031... e1 6/
1 .1813918433423787507... dx
x!1
1 .18143677605944287322...
012
3/1
)13(2
11,
3
4,
3
1,
3
12
kk k
F
...
1
1
!k
kk
k
FF
1 ...
primep
p 31 Berndt 5.28
=
1
3
)(
)6(
)3(
k k
k
Titchmarsh 1.2.7
=
squarefreeq
q 3
...
0
4
22 )(sin
3
2dx
x
x GR 3.852.3
.18174808646696599422... 2 2
124 2 1 2 2
2
24 arcsinh log
log
=1
2 2 121
kk k k( )
...
3
5
2
3log3
322
3
... 5
cot2
25)3(10log25
6
5125
2
5
2sinlog
5
4cos
5sinlog
5
2cos50
=
k
k
k
k
kk 5
)3()1(
5
1 1
134
...
123
2
34
1
27
16
9
2
4
2log3
18 k kk
= 11
2
... 612
6 2 6 33 1
3 1
1
6
2
3 21
log log( )k
k k k
... ( )k
kk 31
... HypPFQk
kk
1
211
1
4
3
4
1
16
14
20
, , , , ,
.18214511576348082181... 2 2
1
1122
2
21
1
loglog
( )( )k
k
k
k
...
212 3
)3(
)13(3
1
13
1
kk
k
kkk
kk
.182321556793954626212...
11 6
10,1,
6
1
6
1
5
6log
kk k
Li
=
0
12 )12(11
1211arctanh2
kk k
J248
.182377638546170138737... 1
42 2
3
2
1
8
12
22 1
1
1
Fk
k
k
k
kk
, , ,( )
... 11
6 4
2
23 1 3 2
1
32
log( ) ( )
( )k
kk
... ( )
1
3
1
31
k
k k
... ( ) ( ) ( )( ) ( )3 11
42 22 2 i i
=x x
e edxx x
2 2
0 1
cos
( )
...
14
12
2 log
)3(
1
9
)2(
54 xx
x
kk
... e10
.18316463254842946724... 1
41 2 1
2
1
log( cos )sin cos
k k
kk
... 5
2 22 2
2
27 41 4 1 4
42
/
/ /cot cothkk
= 2 4 11
k
k
k ( )
... ( ) ( )
arctan
1 1
2 1
1 1 1
2 2
k
k k
k
k k k
k
.183392750954659037507...
0
)1(
22
1
2
1
k
k
k
...
2
1
2
1
4
1
2
3
4
1
2 10
, ,( )
k
k k
= dx
xlogcot
/
0
4
GR 3.511.
... G
2
.1835034190722739673... 12
3
1 2 1
2 2
1
1
( ) ( )!!
( )!
k
kk
k
k
... dxee
xxx
0 1
sin1
2
coth
.183578490978106856292...
12
)()1(
2
1
14
)2(
12
)2(
11k
k
kk
kk
kkk
.18359374976716935640... ( )k
k
k 221
...
...
11 14
)2(
12
)2(
kk
kk
kk
...
3
1)3(
... e e 2cosh
...
1 )!12(2
1
k k
k
e
=
1
2xe
dxx
= 3
15
12
1sinh
2
11sinh
x
dx
xx
dx
x
...
12 644
1
6
1
2
5tanh
20
5
k kk
=
1
2
4125
148
kk
k kF
... 1
3 2 1 22 k kk
/ /
... 1
3 22 k kk
/
...
primep p 31
1log)3(log HW Sec. 17.7
=
23 log
)(
k kk
k Titchmarsh 1.1.9
3 .18416722563191043326...
121k k
k
...
1
16 443
5log
xx
dx
... 3
4
3 3
42
6 1
12
121
log ( )!
( )!( )
k
k kk
.18432034259551909517...
012
1
)4)(1(
)1(
3
)1(
18
5
3
2log2
k
kk
kkkk
= 4
1
1
0
2 11log)1log(
x
dx
xdxxx
... 3
32
20
27
2 1
4
4 4 1
4 1
2 2
1
2 2
2 32
k k k k
kkk k
( ( ) ) ( )
( )
...
00
4/1
2!)!12(
1
)!12(
!
2
1
kk
k kk
kerfe
= dxe
ex
x
1
0
2
...
21
121 2sin
1
)!12(
)1)2((2)1(
kk
kk
kkk
k
... )6()3( 1 .184860269128474... 1 6( )
.184904842471782293011...
2
/1
1
12
11
!
)1)12(()1(
k
k
k
k
ekk
k
...
1 1
2
2
/11
1
kxk
dxe
x
e
k
eee
... 8
491
8
7 81
logkHk
kk
... 1 3 2( / )
... sin1
31
3x
dx
x
8 .1853527718724499700... 67
...
1 !
)(1k
kk
e
k
HeeEie
...
1
164
2
48
2
12
1
16
1
43
2 3
3Li Li( )log log
=log2
31 4
x
x x
...
111112 2
)(
4
1
22
1
14
2
12
1
kk
kkk
kk
k
kk
k
... 256103
2610003
5 H ( )
...
32 22 2 2
2 12
2
2 21
csc sin( )
k
kk
... 4/
0
tan8
2log
2
dxxx
G GR 3.747.6
...
22
1
1 11log
11)12()1(
kk
k
kkk
k
... 1
22 1 3 2 2
221 1
1
log log( )
Hk
k
...
2
/3
2
1!
)1)((3
k
k
k
k
ek
k
.18618296110159529376... ( )
1
2 20
k
kk
... 23
3
12
2 11
k
kk k
k( )
= dxx
x
1
031
1log
...
0
4
3 )!34(2
sinh
k
k
k
... H
kk
kk
( )2
1 6
... 8
6 21
2 4
2 3
1 3log log
arcsinh
= arcsinxx
xdx
1 2 20
1
GR 4.521.3
.18646310636181362051...
( )3
41
kk
k
...
1
12
2
!
2)1(
)2(2
k
k
k
k
k
k
e
I
.18650334233862388596... 1
11 33
2
1
1kk
k
k
k
( ) ( ) 1
=
3
5
6
1
61 3
1 3
21 3
1 3
2
i
ii
i
1 .18650334233862388596... 1
11 1 35 4 3 2
0
1
1k k k k kk
k
k
k
( ) ( ) 1
=
23 12
1
k k
k
=
2
3131
2
3131
6
1
23
ii
ii
= 1
63 2 2 1 1 2 1 2 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /
1 .1866007335148928206... k
ekk
11
.1866931403397504774... 1
21
1
23 4 1
1
2
21 1
1
1
F ( , , ) ( )( )!
k
k k
k
k
... 2
12sin
4)!(
)!2()1(
4
1sin
)2/1cos(2
1
02
k
k
k
kk
k
...
...
1156
2log
36 xx
dx
...
12 2
1
2
7tanh
7 k kk
1 .18683727525826858588...
0
3
3)!1(
1133
kkk
...
1
3
12
)23(
)1(2736)3(6
16
1
k
k
kk
... k
k
k k
k
k
3
2)1(
7
3
7
2
0
...
2 )1)((2
1)1(
kk k
k
...
1 23
1
32
1arcsin11
121
12
11
1
k k
k
k
... 7
K Ex. 108d
.187500000000000000000 =
1
1
3)1(
16
3
kk
k k
=
21
1
)2)(2(
)1(
kk
k
kk GR 0.237.5
=
1 116
4)(
kk
kk
... 12 24
22
2
22
0
1
i i i
Li e x x dxilog ( ) log( cos )
.188016647161420393823...
2 )1)(2(
1)(
k kk
k
... 9
1
... dxx
xG
1
21
)1log(
8
2log
= GR 4.227.13 4/
0
)cot1log(
dxx
...
1/1
2
11
)!1(
)()1(
kk
k
k
kekk
k
.1883873799298847433... 21log22
1
2
1
= 1
42 2 1
1
21
1
2
1
2
arcsinh arctanh
= 1
2 2 1 2 11k
k k k( )( )
1 .18839510577812121626...
0
2
)!2(
42)1(1csc
k
kkk
k
B [Ramanujan] Berndt Ch. 5
...
12
2
)2)(4(6
11
k kk
k
...
24 10
1
k k
.18872300160567799280... 2
)21(82log4)21log(2
=
8
3sinlog
8sinlog22log4
8cot
28
=
k
k
k
k
kk 8
)1()1(
8
1 1
12
Prud. 5.1.5.21
= dxx 1
0
6 )1log(
... 4
91
16
9
1
41
log( )k Hk
kk
.1887703343907271768... 1
2 1( ) e J147
... 3
4 , volume of the unit sphere
3 .1889178875489624223... sinh
2
3 21
22
2
kk
...
12 )!2(!
1
2
1)2(
k kkI
...
1 )1(3
1
3
2log21
kk kk
J149
=
1
1
)1(2
)1(
kk
k
k
... 2 11
4 34
0
( )k
k k
... )1()1()()( iiii ... )2()2( ii
... 1 4
32
0
1
x xsin dx
...
12
/1
2
12
)!2(
)()1(
k
k
k
k
kke
k
k
... dxexe
erf x
1
2 2
2
1
4
1
...
1
1 2
)12(2
)1(
2
2arcsinh21
kk
k
k
k
k
...
2
22 1)(k
kk
... dxxx
xdx
xx
x
2
12
21
0
23 log
)1(
)1(log
3
2log
4
)3(
.189727372555663311541...
122
1 )18(
8
4
)12(
2
11
22
11
28
1
kkk k
kkk
406 .189869040564760332...
0
2
!
cosh
24
1 22
k
ee
k
keee
.1898789722210629262... 1
2
1
1
1
21
csch
2
( )k
k k
...
2 5
517
20
1
5
1
23
csc( )
k
k k
...
12
1 9
11log
9
)2(
33
2log
kkk kk
k
...
22
1 11log
1)1()(
)1(
kk
k
kk
kkk
k
... 13
13log
... 1
1 32
kk
1 .190291666666666666666 = 6)3(
24000
28567H
...
1
21
21
2)!1()!1(
)!()!(124
2
7
kkkk
kkE
.190598923241496942... 15 8 3
6
1
6 2
2
1
kk k
k
k( )
...
2
1)(12
)1(
kk
k
k
...
3
7
...
12 1
2 1
2
20
1
arctan tan sin
sin
x
xdx
... 11
220
( )k
kk
.1906692472681567121...
1 )!2(
1
2
)1(
24
9
k kk
Eie
=1
1 33 ( )! !k kk
...
12
22
66
126log6log5log25log
2
1
kk
k
k
HLi
=
12
1
5
)1(
kk
k
k
... 53
...
22
22
111)(
kk kkLi
k
k
...
12 544
1
5
1
8
tanh
k kk
...
12
1
2
)1(
k
k
k
... 2
8
.19166717873220686446...
( )2
51
kk
k
.19178804830118728496... 4 2
3
2 3
3 41
2log log
dx
x x
... ( )
( )
2 1
3 1
...
11
1 1sin
1
)!12(
)12()1(
kk
k
kkk
k
... G2
... 1ee
e J152
... log k
ekk
1
.1923076923076923076 =
0
5sin
26
5dx
e
xx
=
0
)12/()5(log)1(5logcosh2
1
k
kk e J943
.192315516821184589663... 3
... 3
...
1
3
8kkkH
.19245008972987525484...
0
2
2
)1(
33
1
kk
k
k
kk
.19247498022682262243... 2 6 2 3 62 1
2 31
log log( )
( )!
k
k kkk
!!
... 26
3
...
1 2 11 2
0
e Eix
e xdxx( )
( )
...
4
3
4 4
1 2
2 1
1
0
loglog ( ) log( )
k
k
k
k
1
[Ramanujan] Berndt Ch. 8
...
129
11
3sinh
3
k k
.193147180559945309417...
222 )12(2
1
24
1
2
12log
kk kkkk
=
1 )12(2)12(
1
k kkk J236
=1
8 231 k kk
[Ramanujan] Berndt Ch2, 0.1
=
15
1
0 4
)1(
3
)1(
k
k
k
k
kkk K Ex. 107f
=
2 2
1
kk k
=
0 )3)(2)(1(2
1
kk kkk
J279
=
112
2 2
)12(
2
1)(
kk
kk
kk
=
=
134
223 xx
dx
xx
dx
=dx
x x x
dx
x( )log
11
12
12
13
=
02 sinh)1( xx
dxx
GR 3.522
= dx
e ex x20 1
... 7)3(
8232000
9822481H
... 2
9
32
2
21
k
k k
kk
... e i i / /4 2
...
1
2/1
1
1)12(!
1
kkk
k
ek
...
1
1
)!1(k
ke
k
ee
... )7()3(
... 1log1
!
1)3()1(
3
21
1
k
kk
k ekk
k
...
log log log2
1 3
2
1 3
2
i i
=
3/23/1 )1()1(2
1log
= log ( )( )
11
13 1
32
1
1
k
k
kk
k
k
.194035667163966533285...
0
2 32
)1(
2
3csch
3
6
6
1
k
k
k
... ( ) ( )
log
1 1
1
1
21
1 1
2
1
422
2
2
k
k k
k
k
k
k k
k
...
1
)3(1
4)1(
kk
kk H
... 2
1)1(1)1(1
44/14/1 i
4/34/3 )1(1)1(14
i
=
1
1
24
1)14()1(1 k
k
k
kk
k
...
122 2
1
22
2cot
4
1
k k
.194444444444444444444 =
11 )3)(1(
10,1,
7
1
736
7
kkk kkk
k
... 2
1
... e k
k
k
k
2
1
3
2
2
2
( )!
... e
k
k
k
2
0
1
2
2
1
( )!
... e k
k
k
k
2
1
1
2
2
1
( )!
... 2 2 22 1
12
0 11
e I Ik
k kk
( ) ( )( )
!
.194700195767851217061... 1
41
41 41
1e
k
kk
kk
/ ( )!
... 9
... 2
1
2
1
2 2 10
erfik kk
k
! ( )
... 16
sin
... 4/
04
3
cos
sin
3
22
dxx
x
...
12
24
2
3
)12()2(
3
11
3cot
3)2(
kk
k
kk
kk
k
...
1
1)4(2
k
kk
k
...
1 )!1(k
k
k
kB
... dxe
xHypPFQ
x
1
222 log
1},2,2,2{},1,1,1{26
... 2
cot2
...
1
123
3
72
3
72
1
12
1
33
2 3
3Li Li( )log log
=log2
31 3
x
x xdx
....
2
2)1)()1(
k
k
k
k
... ek
k kk
(cos )/ cossin cos
!1 2
0
1
2 2
GR 1.463.1
= 2/2/
2
1 ii ee ee
...
0
2
6
1133
kk k
k
5 .19615242270663188058... 27
....
1
2
1
13
)1(
3csch
322
1
k
k
k
...
1
2
92
2log3
8
3log9
kk
kH
...
21
2
0
tanh sinsinh
x dx
x
.19634954084936207740...
16
=
04 4)12()12(
)1(
k
k
kk K Ex. 107e
= ( )
( )sin
1
2 1
2 1
4
1
21
k
k k
k J523
= sin cos sin cos3
1
3 2
1
k k
k
k k
kk k
=
22
2
032
2
)4()1( x
dxx
x
dxx
=
044 xx ee
dx
1 .1964255215679256229... H kkk
21
2 1( ( ) )
... 44
2 2
20
1
Gx dx
x arcsin
= dxx
xx
2/
02
2
sin
cos
GR 3.837.5
= dxxx
x
022
3
1
arctan GR 4.534
... 14},1,1{,2
12
)!(
1
12
HypPFQk
k
kk
...
3
1
6
3log
318327
2
9
)3(13 )1(3
=
12)13(k
k
k
H
...
0
2
10
!
2)1(22
k
kk
kJ
... e
e
k
e
k
kk( )
( )
1
12
1
1
.1967948799868928... ( )( )
15 1
1
1
kk
k
k
... 2
3
... 23 3
2 1
2
0
( )
)
x
e edxx x
=
023
2
123 xx ee
dxx
xx
dx
... 9
... ( )
! ( )
1
2
k
k k k
... ( )
! ( )
1
2
k
k k k
... 25 1
32
27 1
32
13
16
1
32 2
5
1
cosh sinh
( )
e
e
k
kk !
... 3log2
... 12
2
1 4
2 1 2 1
1
1
si
k k
k k
k
( ) ( )
( )!( )
...
12
)1(2
)4/1(
1
4
18
k kG
.19739555984988075837...
0
12 )12(5
)1(
5
1arctan
kk
k
k
=
1
2)2(2
1arctan
k k [Ramanujan] Berndt Ch. 2, Eq. 7.6
...
0
12 )12(5
)1(
5
1arctan
kk
k
k
...
1
2
)2)12()2((128
3
k
kkk
.19749121692001552262... 2 13 1
21
2 11
2
1
cossin
( )( )!( )
k
k
k
k k
...
12 116
21
4coth
4 k k
...
11 9
)2(
)13)(13(
1
362
1
kk
k
k
kk
... 1
254 3 2 2 5 4 2 1
41 2
1
arccot log log ( )k kk
k
kH
... 60 222 101
ek
k kk !( )
.19783868142621756223... 2 2 2
04
2
2
2
4 48
log log log sinx x
xdx
2
... 19
12 327
13 3
8
1
16
7
6
1
1 3
32
1
31
( ) ( )
( /( )
k
k k )
.19809855862344313982... 11 5 23 2
2
1
1k kk
k
k
k
( ) ( ) 1
...
( )log
33
1
k
kk
...
4
54
3
2)1()1(2
KE
= dxxdxx
x
2/
002
2
sinsin1
sin
...
1
2)12()2(k
kk
...
0
21 )32()!)!12((
)1(2)1(
k
k
kkH
...
1
21
2
)1)1(()1(
kk
k k
...
1
4
)!1(
4
4
13
k
k
k
ke
...
0 23
)1(
kk
k
...
4 32 3
17
66
1
1224
csc( )k
k k
... ( )
! !!
1
0
k
k k k
... 1
0
2 log)1(2223 dxxxeEie x
... ( )( )
log ( )
1 21
1
k
k
k
kk
1
...
1 2log
k
k
kk
...
1 )74(
1
7
8log
14147
40
k kk
... 8
1
...
3
2
3
13log3log2log
6 222
2
LiLi
... 2
22
1
2
2
0sintan
kk
k
k
Berndt ch. 31
... 4
21
22 2
1
21 1cos ( ) sin ( ) cos CosIntegral SinIntegral
= dx
e xx 20 4
...
13
1
2
)1(
k
k
k
k
... root of ( ) ( )x e x ex x 1 1
... 3
.20000000000000000000 = 1
5
1
6 5 1
2
1 1 0
4
1
kk
kk
x
ek x dx
e
x dx
x
( ) cos log
= k k
k kk
( )
( )( )
5
4 11
J1061
.20007862897291959697... k
kk
k
k
k
3
62
1
111 6 3
1
( ) ( ( ) )
=
2
34)31()2()2(
6
1 iiii
2
34)31(
2
34)31(
ii
ii
2
34)31(
ii
.2000937253541084694... 3
2 2 31
3
1
21
log( )k
k k k
=2
13
1
0
3 11log)1log(
x
dx
xdxx
.2000937253541084694... 3
2 2 11
30
1
log log
xdx
.2002462199990232558... 2 2 2 1 2 21
2 2 11
log( ) log( )k
k k k
= 2 2 2 1 2 arcsinh log
.2003428171932657142... ( )3
6
1 .2004217548761414261... 1 1 1
¼ ¾ ¼ ! ¾ !
.2006138946204895637... 1 2
2
1
81
21
2
3
2 2
3
2 2
log ii
i i ii
i
= 1
8
1
2 2
1
2 21
21
2 2
i i i itanh
3
4
2
2 8 2
logcoth
= ( )
1
1
1
3 21
k
k k k k
.2006514648222101678... Ei Ei( ) ( ) e e e x
10
1
dx
...
0
2
)1()!2(
4)1(1sin2sin
k
kk
kk
6 .201255336059964035095... 3
5
.2013679876336687216... 9
8
2
3
2
0
e k
k
k
k ( )!
.20143689688535348132... ( )( ( ) )
!
11 2
2
k
k
k
k
.2015139603997051899... 1
6
6
3
3
2
1
2 320
csch ( )k
k k
1 .20172660598984245326... 1
3 21k
k
.20183043811808978382... e
e
k
kk8
3
8 2
2
1
1
( )!
... e
e e
k
ek
kk
sin
cos( )
sin1
1 2 112
1
1
.20196906525816894893... 16 4 3 5 8 12
12
( ) ( )
i i
=1
41
2 5
47 51
1
1k k
k
k
kk
k
( )( )
... 1
2 1G
.2020569031595942854... ( )log
3 11 1
2 132
2
0
1
k
x x
xdx
k
GR 4.26.12
= 6 2
1 1
3
2 30
2
x x
x
dx
e x
( ) Henrici, p. 274
=
1
0
1
0
1
0 1dzdydx
xyz
xyz
1 .2020569031595942854...
1
32
2
13 )1(
1331)3(
kk kk
kk
k
= )1,2(MHS
= H
kk
k ( + )1 21
Berndt 9.9.4
=
122 )1(
)12(
k
k
kk
kH
=
1
)2(
)1(k
k
kk
H
= 5
2
12
5
2
1
2
1
31
1 2
31
( ) ( ) ( !)
( )!
k
k
k
kk
kk
k
k k Croft/Guy F17
Originally Apéry, Asterisque, Soc. Math. de France 61 (1979) 11-13
= ( )( !) ( )
(( )!)
1205 250 77
64 2 11
10 2
5k k k k
k Elec. J. Comb. 4(2) (1997)
=
1
23
3
123
3 )(2
180
7
1
12
180
7
kk
kk e
k
ek
Grosswald
Nachr. der Akad. Wiss. Göttingen, Math. Phys. Kl. II (1970) 9-13
=
2
12
)(4
45
2 22
123
3
kke
k
kk
Terras
Acta Arith. XXIX (1976) 181-189
= 4
7
2
4 2 2
2
72
7
2
0
2 2 ( )log
k
kkk
Yue 1993
= 4
9
2
4 2 3
2
92
2
27
2
0
2 2 ( )log
k
kkk
Yue 1993
=
22
4 2 2 2 32
0
( )
)(
k
k kkk
Yue 1993
=
4
7
2
4 2 1 2 2
2
0
( )
( )(
k
k kkk )
Ewell, AMM 97 (1990) 209-210
=
8
9
2
4 2 1 2 3
2
0
( )
( )(
k
k kkk )
Yue 1993
=
0
2
)32)(22)(12(4
)2()52(
3 kk kkk
kk Cvijović 1997
=
0
22
)32)(22)(12(4
))2(1)(52(
32
3log31
3
4
kk kkk
kk
Cvijović 1997
=
0
122
)22)(12(16
)2()12(4
4log
2
3
14 kk
k
kkk
k
Ewell, Rocky Mtn. J. Math. 25, 3(1995) 1003-1012
= primep
p 41)6(
=
primep pp
ppp21
321
1
1)4(
= 5
41 1 1 1
3
22 2
1
4HypPFQ[{ , , , },{ , , }, ]
= dxx
x
1
0
2
1
log
2
1 GR 4.26.12
= log( ) log10
1
x xdx
x GR 4.315.3
= 4
32
0
4
log(cos ) tan/
x x d
x GR 4.391.4
= 16
32
0
1
arctan log (log )x x x d x GR 4.594
= x dx
ex
5
0
2
1
= 2
3 1
3e
edx
x
e x
GR 3.333.2
= 1
2 1
3e
edx
x
e x
GR 3.333.1
2 .2020569031595942854... ( ) ( ( ) ( ))( )
3 1 22 1
12
2
32
k k kk k
k kk k
... )1)1316((1
12133
kk
kkk
... )1)1215((1
12143
kk
kkk
.2020626180169407781... ( )log ( )
12
k
k
k
k
... )1)1114((1
12113
kk
kkk
... )1)1013((1
12103
kk
kkk
... )1)912((1
1293
kk
kkk
.20211818111271553567... )1)811((1
1283
kk
kkk
.20217973584362803956... )1)710((1
1273
kk
kkk
.20218183904523934454... ( )( )
!
12
12
2
2
1
kk
k
k
k
k
ke
k
.20230346757903910324... )1)69((1
1263
kk
kkk
.2025530074563600768... 4/34/1 )1(cot)1(cot82416
7
4
ii
4/34/1 )1()1(4
)2()2(8
1 i
ii
= 1
8 5 13 5
2 1k kk
k k
( ( ) )
.2026055828608337918... ( , , ) ( )1
54 0
1
5
1
54 41
Likk
k
.20261336470055239403... 3 1
81
2
3
1
cos( )
( )
!
k
k
k
k
.20264236728467554289... 2 2
22 21
( )
( )
k
kk
= x x dx x x dxcos cos 0
12
0
1
.20273255405408219099... log log3 2
2
1
5
arctanh AS 4.5.64, J941, K148
= 1
5 2 1
1
2
1
3
1
22 10 1
1
11
kk
kk
k
kkk k
( )
( )
k
=
1 1
)(
1
1
32
1
3 n kk
kn
kkk HH
= log
( )
x dx
x2 1 21
= ( ) sin sinx x x
xdx
5
0
1
GR 1.513.7
.2030591755625688533... 1
7 4 13 4
2 1k kk
k k
( ( ) )
... ( )
log( )6 3 1
11
3 6
2
k
kk k
k k
3 .203171468376931... root of ( )x 1
... tan(log )
i ii i
i i
.2032809514312953715... log3
4
.2034004007029468657... 1 11
1
1
1
1
21
1
2
Eik k k k
k
k
k
k
( )( )
!( )
( )
( )! !
.2034098247917199633... ( ) ( ) ( )2 3 3 2 4
.2034498410585544626... 3 3
12
1
2 3 20
( )
( )(
k
)k k k
.20351594808527823401... log ( ) 31
kk
.2037037037037037037 = 11
54
1
3 91
k kk ( )
... ( ) ( )( )
2 2 326
27 2 32
k
kk
.20387066303430163678... 1024 12832
3
2
45768 2 16 3 5
2 4
log ( ) ( )
=1
41
5
46 51
1
1k k
k
k
kk
k
( )( )
61 .20393695705629065255... 5
5
.2040552253015036112... 1
51
2
53 21
1
1k k
k
k
kk
k
( )( )
.2040955659954144905... ( ) ( ) coth4 4 2 4
28
=1
41
2 4
46 41
1
1k k
k
k
kk
k
( )( )
.20409636872394546218...
3
1
12
1
6
3 3
2
3 3
2
i i
=1
6 3 13 3
2 1k kk
k k
( ( ) )
... ( )
log( )5 2 1
11
2 5
2
k
kk k
k k
.20420112391462942986... ( )4 1 1 1
21
22
4
k
kk Li
kk k
.20438826329796590155... H k
kk
( )3
1 6
... loglog
2
2
[Ramanujan] Berndt ch. 22
... e
x e dxx4
2 1
0
2
.20487993766538552923... 2 1 2 2 2 2 2 32 1
4 1
1
1
log log( )
( )
k
k k k
k
kk
.205324195733310319... ( , , )1
53 0
1
5
1
53 31
Likk
k
.205616758356028305...
2 1
2 21148
2
8
1
42
( ) ( ) cosk
kk k
k
k
= )()(2
122 iLiiLi
=log x dx
x x31
=
x x d
x
log2
0
1
1
x
= log 11
41
x
dx
x
.20569925553652392085... ( )
1
2 1
1
1
k
kk
.2057408843136237278... 11
61
kk
.2057996486783263402... 7
180
3
31
coth k
kk
...
0
33/5
sin3
4
2
13 3
dxxe x
...
2
3
1
3 32
2
20
1
Lix
xdx
log
( )
.2061001222524821839... 2
2136
49
720
1
6
k kk ( )
.2061649072311420998... ( )kk
2
2
1 1
.2062609130638129393... 15 2 13 2
2 12k k
k ik k
( ( ) ) Re ( ) Li
.20627022192421495449... ( )3 2
6
1
615 2
1
k
k kkk k
.2063431723054151902... 2 2 2 4 2 3 2 3 41
32 2
2 log log log log Li
=( )
( )
1
2 2 20
k
kk k
.20640000777321563927... ( )( )
13
5
1
5 11
13
1
kk
k k
k
k
... ( )
log( )4 1 1
11
4
2
k
kk k
k k
.2064267754028415969... 3
6 3
3
4
1
3 122
coth
kk
= ( )( )
12 1
31
1
kk
k
k
.2064919501357334736... H kk k
k
kkk k
( ( ) ) log
3 11
113 2
2
... 1
2
3
21
1 3
22
log ( )( )
k
k
k
k
Srivastava
J. Math. Anal. & Applics. 134 (1988) 129-140
.2066325441561950286... ( )3 1 1
21
2 32
k
kLi
kk k
.2070019212239866979... I
e
k
k kk
k
k
04
0
41
2 2( )( )
!
1 .20702166335531798225... 1 11
2 11
e erfik kk
( )!( )
=e
xdx
x 2
12
0
1 GR 3.466.3
5 .2070620835894383024... e
e
e
k
e k
k
1
10 ( )!
4 .2071991610585799989... 7 3
2
1
1 1
2
0
2
1
0 ( )
sinh
log ( )
( ) sinh(log( ))
x dx
x
x
x xdx
1 .207248417124939789...
1
4
1
2 10
eerfi e e erfi
e
k
k kk
( )sinh
!( )
.2073855510286739853... ( )5
5
.20747371684548153956... ( )( )
( )
12 1
2 11
1
k
k
k
k k
.20749259869231265718... 10
8
754
0
1
x xarcsin dx GR 4.523.1
... 1
22 2 1
12
log ( )( )
( )
k
k
k
k k
2 .2077177285358922614...
2
23 ))()((k
kk
.2078795763507619085... iii eei log2/ )valueprincipal(
=
2sinh
2cosh)sin(log)cos(log
iii
.20788622497735456602...
1
2
3 .207897324931068989... e e e
e
k
k
e e
k
2 1 2
0
1
2 1
/ cosh
( )!
1 .208150778890... 1
42 ( )kk
.2082601377261734183... 3
32 2 22 3
dx
x( )
.2082642730272670724... 256)3(42log192)4(3
832
2
=1
41
4
45 41
1
1k k
k
k
kk
k
( )( )
=5
24
15
1
log( )x
xdx
.2084273952291510642... e Ei Eie
x
edxx( ( ) ( ))
log log( )
2 1
2 1
0
1
.20853542049582140151... 2 2
3482
2
48
1
8
1
2
1
82log
log
Li Li 3 LHS WRONG
=log2
31 2
x
x xdx
.2087121525220799967... 5 21
2
1
7 1
2
1
kk k
k
k( )
= 11
3 1
21
0
( )
( )
k
kk k
k
k
1 .2087325662825996745... e ek e
ek
k
1 1
0
1
1
/
( )!
.20873967247130024435...
( ) Re( )
( )3 4 2 1
22 1
2
3
21
i i k
i kk
=1
41
2 3
45 31
1
1k k
k
k
kk
k
( )( )
6 .208758035711110196... e dxxsin
0
.2087613945440038371... 2 1
2112
2 2 21
1
log( )
( )
k
k k k J365
= ( )
Re( )
( )
1 3
23 22 1
k
kk
kk k
k
i
= GR 4.221.2 log log( )x x10
1
dx
1 .2087613945440038371... 2
0
1
122 2 1
1 1
log
( ) log( )x x
xdx
2 .2087613945440038371... 2
2
1122 2
1
log( )
H
k kk
k
=
log log1
1
0
1
xx dx
= log( )log1
21
x x
xdx
1 .20888446534786304666... ( ( ) ) k k
k
1 2
2
... ii ee 2log2log2
11cos)1(
=
2
1)(cos
k
kk
k
.20901547528998000945... 6
5
1
6
1
5
1
5
1
6 1 62 21
2
1
Lik
Hk
k
k
kk
( )
( )
.2091083022560555467... ( )( )
14 1
4
1
415
1
kk
k k
k
k k
=
1
2
1
2
3
2
3
2
i i
.2091146336814109663... 2
8 2
1
3
1
4 421
tanh 3
k kk
.2091867376129094839... ( )
( )!sinh
2 1
2 1
1 1
1 1
k
k kk k
k
.2091995761561452337... 2
3 31
1
2
2
0
2
( sin )
sin
/ x
xdx
1 .2091995761561452337... 2
3 3 2 1 2 2 10
2
0
k
k
k
kkk k
!
( )!!
( !)
( )! CFG D11, J261, J265
= 1
22 1
0 k
kk
k
( )
= 9
3 1 3 1
2
1
k
k kk ( )( )
=dx
x
dx
e e
dx
xx x30 0 01 1 2
sin
=x dx
x30 1
GR 2.145.3
= dx
x x
dx
x x20
211 1
... = 2 2
20
2
4 2 2 1
log log
x dx
x
.20943951023931954923...
6
2)cos(sin
15 x
xxx Prud. 2.5.29.24
.20947938452318754957...
2 12 2
1
2
)(
2
12log
k jkj
kk j
kk
k
.2095238095238095238 = 22
105
1
2 1 2 9
1
4 2 4
2
0 0
( )( ) ( )( )k k k k
k
kkk
k
.209657040241010873498... log( )csc ( )
1 2
1
k
kkk
1
.2099125364431486880... 72
343
1
82 0
1
8 82
2
1
( , , ) Lik
kk
.2099703838980124359... ( )log
!
11
k
k
k
k
2 .2100595293751996419... 10
81 3 1 1
3 2
20
1 2
3
2
300
log log logx dx
x x
x dx
x
x x d
x 1x GR 4.261.2
= log2
2 10
x
x xdx
1 .2100728623371011609...
1
02
2
12
log
2
1,2,
2
1
8
1
8
2
x
dxx
...
1
31
)()3(log
)()3(
kk k
kk
k
k Titchmarsh 1.6.2
... 4
13
6
1
4121
( )!
( )!
k
k kk
Dingle p. 70
.2103026500837349292... 1
51 ( )k kk
.21036774620197412666... sin ( )
( )!
( )
( )!( )
1
4
1
4 2 1
1
2 1 2 10
2
1
k
k
k
kk
k
k k
2 .2104061805415171122... ( ) ( )3 7
.210526315789473684 =4
19
.2105684264267640696... 6 3 220
9
1
2 10
(log log )( )
( )(
4)
k
kk k k
8 .2105966657212352544...
2
3( )
.21065725122580698811...
12
2
6
1
6
1
2 1 2 40
log ( )
( )( )
k
k k k
= x x2
0
1
arctan dx
.21070685294285255947... 1
3
1
3
13 7
1
e x
dx
xcosh
... log( )10
e kk
k
... 11
21
( )k
k k
46 .21079108380376900112... 17e
.2109329927620049189... 1
82 2 4 ( ) ( )i i 1
=
2
3
8
1
42 2
142
( ) ( )i ik
kk
= )()(4
1
8
1
2ii
= ( ( ) ) 4 1 11
kk
... 2 3 11
2
20
e Eix
e xdxx( )
( )
.21100377542970477... ( , , ) ( )( )1
52 0
1
5
1
5
1
42 21
1
1
Lik
k
kk
kkk
H
k
= ( ) log log log2 4 5 54
52
2
Li
2 .211047296015498988... 2 2 1
1
1log log ( ) /
k k
k
1 .2110560275684595248... E(¾)
.21107368019578121657... ( )
!/3 1
11
1
2
3k
ke
k
k
k
7 .2111025509279785862... 52 2 13
...
11
2 1
21
( )
( )!!
k
k kk
!! J166
.2113553412158423477...
( )3
21
kk
k
.2113921675492335697... 1
21( ) Berndt 7.14.2
.2114662504455634405... ( )log
3 1 11
13
21
3
2
k
k kmm
kk
mk m
=
log cosh log logcosh
1
3
3
23
3
2
=
2
35log
2
35log
ii
.2116554125653740016...
e k
k
e
k2 1
sin
.2116629762657094129... 14
coth
8 .21168165538361560419... Ei e( )
... 2 4 32
3
23
3
162 2
4
7 3
4
2 3 2
GGlog
( )
= H
kk
k ( )4 1 21
... ( )
!
!
1
23
k
k k
.21184411114622914200... 8 216
3
1
2 2 80
log( )
kk k
... 1
42 2
0
1
log ( ) log sin cosci x x x dx
.2118668325155665206... 2 1 11
11
e e ee k kk
k
( ) log( )( )
J149
.21192220832860146172... 3 1
318
41
1 2
Gk
k
k
k
( )
( / )
1 .21218980184258542413... 1
3 1 30k
k
/
.21220659078919378103... 2
3
0
3 23
0
1
/
sinx x dx
.21231792754821907256... log log31
22 2 3 2
1
2
dx
x x
... 6 32
3
0
( )
x dx
e ex x
=
0 13/1xe
dx
.2123457762393784225... e
.21236601338915846716... ( )2 1 1 1 1
21
22
k
k kLi
kkk
k
.21243489082496738756...
( )k kk
k 2 11
.2125841657938186422...
2
2
1 42 20
2e
x dx
x
x dx
x
cos cos
AS 4.3.146, Seaborn Ex. 8.3
= sin( GR 3.716.6 tan ) tan/
20
2
x x d
x
= cos( tan ) sin20
x xdx
x
GR 3.881.4
.2125900290236663752... log k
kk3
1 1
.2126941666417438848... log log2 21
2 42
31
H
k kk
k
.21271556360953137436... 10 3 24 2
108
1
3
2 1
3 21
log ( )k
k k k
.2127399592398526553... Ik kk
3 0 10
21
64 1
1
3( ) (; ; )
!( )!
F LY 6.114
.21299012788134175955... 2 2 6 4 21 2 1
2 2
1
1
log log( ) ( )!!
( )!
k
kk
k
k k
=
1
1 2
8
)1(
kk
k
k
k
k
.2130254214972640800... ( ( ) ( )) k kk
1 2
2
.2130613194252668472... 2
11
2 1
1
1e k
k
kk
( )
( )!
.2131391994087528955...
02 )22)(12(4
)2(
4
)3(7
kk kk
k
Yue & Williams, Rocky Mtn. J. Math. 24, 4(1993)
...
8
1
2
3
29 42 2
0
2
ee erfi erfi e x x dxx
/ sin cos
.2132354226771896621... hgk k
k
k
kk
k
( ) ( )( )1
7
1
71
1
721
1
1
.21324361862292308081... 1
7
1 .21339030483058352767... 1
22
1
2 111
1k
kk
k
½
3 .21339030483058352767...
111
0
2
)(
2
)(
kk
kk
kDk,
1
0 )()(k
knD
.21339538425779600797... 4
9
2
31 2
0
1
log
logx x x dx
x
GR 4.241.10
GR 4.384.11 log(sin ) sin cos/
x x x d2
0
2
.2134900423278414108...
sin ( )
!( )
4
2
1 41
120
k k
k k k
.21355314682832332797... 2
3
1879
7560
1
3 3
1
1
log ( )
( /
k
k k k )
.21359992335049570832... 1
82 2 2 2
1 2
2 1
1 2
1
( cos sin )( )
( )!
k k
k
k
k
=
1
211
)!2(
2)1(
k
kk
k
k
.2136245483189690209... 1
6
3
2 3
1
32 21
cot
kk
.2136285768839217172...
111 )92(!
1,2
13,
2
11
11
1
k kk
kF
511
32
945
641
eerfi
.2137995823051770829... 1
9
8
27
1
2 2
1
22
1
1
arcsinh( )k
kk k
k
.21395716453136275079... 1
6
3 1
641 1k k
k
kk
k
( )
.2140332924621754206... 5 5 15
515
2
k
k k
kk
( ( ) )
1 .2140948960494351644...
233/23/1
11
))1(())1((2
1
2
3cosh
2
1
k k
.2140972656978841028... 1
1
1 1
1 1e e e
dx
ekk
x
1 .214143334168883841559...
( ) ( ) ( )k k kk
12
.2142857142857142857 =3
14
1 .2143665571615205518... ( )
log
k
kk
1
2
.2146010386577196351...
1 2
2
1
8
3
2
1
2
3
2
1
2
1
2
1
2
log
=( )
1
232
k
k k k
.21460183660255169038... 14
1
2 30
( )k
k k
=( )!!
( )! ( )
2 1
2 4 121
k
k kk
J387
=dx
x x
x dx
x
x
xdx4 2
1
2
20
4
20
1
1
sin
cos
arcsin
( )
/
2 .21473404859284429825... 3
14
1 .2147753992090018159... log log log ( )2 3 3 33
4
x dx GR 6.441.1
.2148028473931230436... ( )
log2
81
1
812
1
k
k kkk k
.21485158948632195387... 8 4 8 5 21
4 2 40
log log( )
( )
k
kk k
.21506864959361451798... ( ( ) )
!
k
kk
1 2
2
.215232513719719453473... ( )
1
2 1
1k
pp prime
, where p is the kth prime
.2153925084666954317... ( )( ( ) )k kk
2
2 1
... dxx
xdxxx
4/
02
34/
0
2
tan
sincossin
26
1
3
1
.2156418779165612598... 24 62
22 2
0
1
arcsin arccosx x dx
... 2
32
7 3
4
4
27
4
2 1
( )
( )
k
kk
= ( ) ( )( )
1 11
22
kk
k
k kk
.2157399188112040541... 7 331 5
4 6 1 2
4 2
51
( )
( )
( /
)
k
kk
.2157615543388356956... 3
4
4
31
31
1
log ( )
k kk
k
H
1 .2158542037080532573... 12 2
22
( )
2 .21587501645954320319...
( )
( )
k
kk
1
1 12
2
.21590570178361070... 1
2
1
2 2
1
0
e e
k e
k
k
( )
... ( ) ( )k kk
1 11
.216166179190846827...
1
12
)!2(
2)1(
4
1sinh
4
1
k
kk
ke
e
1 .2163163809651677137... cos sin sin( )
( )
1
4
1
41
1
21
1
2 10
k
k
=( )
( )!!
( )
( )!!
1
4 4
1
4 2 2 40 0
k
kk
k
kkk k
13 .2163690571131573733... k kk
4 2
2
1( ( ) )
.2163953243244931459... 3 3 3 2 11
3 11
log log( )
kk k
GR 1.513.5
= ( )
( )
1
2 11
k
kk k k
= log 120
1
x
2 .21643971276034212127... k
k kk k
2
21
21
3
21
1
2
.2167432468349777594... 2
323 2 2
1
8 7 8 31
log( )( )(k kk )
J266
1 .2167459561582441825... log( )(
26
1
4 3
1
1 4 121 0
)
k k k kk k
... 1
2
2
2
1
8
1
2
1
2
log i i i i
= cos
)
2
0 1
x
e edxx x
.216872352716516197139... 5
4log
5
4
2
5log5log2log25log2log2)3( 2
322
Li
5
4
5
133 LiLi
= H
kk
kk 5 2
1
.2169542943774763694...
sin
2
21
22
2 k
... = 93 5
2
4
0
( )
sinh
x dx
x
.21740110027233965471... 2
21
9
4 1 3
k
k kk ( )( )
1 .2174856795003257177... 24 2 5 1 241
44 0 10
Ik kk
( ) (; ; )!( )!
F
.2178566498449504... ( )
1
432
k
k k
.218086336152111299701... 1
2 242
1
2
2 1
1
log
( )
( )
kk
k
H
k k
2 .2181595437576882231... 2
5
1 .21816975871035137... j g2 22 J311
.2182818284590452354... ek k k kk k
5
2
1
2
1
2 41 0( )! !( )( )
1 .218282905017277621... arctan( )
( )e
e
k
k k
k
1
2 1
2 1
0
.2185147401709677798... ( ) log3 48 22
38 64
1
4
2
4 31
k kk
= ( )( )
13
41
1
kk
k
k
37 .2185600000000000000 =116308
3125 4
2
1
F kkk
k
3
1 .2186792518257414537... 2 2
2 2
sinh
2 .21869835298393806209...
2
1
1arcsin
kk
...
12
11log
1arctan
21
)12(
)()1(
kk
k
kkkkkk
k
.2187858681527455514... Hk
kk 6
6
5
6
51
log
.2188758248682007492... 2 5log
... 136
625
4 2
625 51
2
4
1
arccsch( )k
k
kk
k
... cosh( ) ( ) sinh
1
2 81 1
12
14
erf erfix
dx
x
.219235991877121... ( )
1 1
31
k
k k k
1 .21925158248756821... 8 2 4 2 2 32 1
22
1
log log ( )( )
H
k kk
k
.2193839343955202737...
Eik k
k
k
( )( )
!1
1
1
LY 6.423
= logdx
e
e
xdxx
x
1 0
1 1
= dx
ee x
0
2 .2193999651440434251... ( ) ( )3 6
.2194513564886152673...
12 2
1
2
11
kkLi
k
9 .2195444572928873100... 85
.21955691692893092525... 2 22
1
4 4 21
( ) coth
k kk
= ( )( )
12 2
41
1
kk
k
k
=
Re( )
( )
k
i kk
2
21
.21972115565045840685... e
e x
dx
x x
dx
x4
5
4
1 1
2
12 7
14
1
cosh cosh
.2197492662513467923... ( )( )
!
1 1 0
1
k
k
k
k
.2198859521312955567... 1 1 10
1
ci x x x dx( ) cos log cos
.219897978495778103... ( ( ) ) kk
2
1
2 1
2265 .21992083594050635655... 3
4
7
.2199376456133308016... ( ) ( )
1
4
1 3
1
kk
kk
H
k
... 3 2 2 2 2 3
6 24
2
2
2
24
2
2
2
6
3
3
log log log log ( )
= log sin2 2
0
x x
xdx
.22002529287246675798... 1
21
1
41
1
21
Jk
k
k
kk
( )( )
( !)
... log21
41
2 31
2 3
i i
1
4
3 3
6
3 3
6
i i
= ( )
1
3
1
31
k
k k k
.2200507449966154983... 2 24 2 1
2
1
2
2
1
log( )!!
( )!!
G k
k kk J385
... ( )
log2 1
21
1
2
1
222
1
k
kk
k kkk k
.2203297599887572963... 2 4 2
1
2 4 2
1
2coth coth tanh
=1
4 4 2
2
221 1k k
k
ikk
k
Re( )
( )
= ( )( )
14
4
1
4 114
1
kk
k k
k
k
1 .2204070660909404377... ( ) ( )3 4
.2204359059283144034... 27
4
3
1
6
1 2)1(
= 3/22
3/12
3/22
)1(3
1)1(
2
)1(
108
5
LiLi
3/22
3/2
)1(4
)1(
Li
= 1
36
1
3
5
6
1
3 2 20
( )
( )
k
k k
=
02
1
03
13 1
log
1
logxx ee
dxxdx
x
xx
x
dxx
...
02 12
3
28
33/2xe
dx
.22056218167484024814... 1
22
1
22 1
1
2
1
1eEi
kH
k
kk
kk
log( )
!
.2205840407496980887... cos Re
2
2 i
.220608143827399102241... 3 3
24 2
36 1
22
0
1 ( )log log( ) log x x dx
...
( ) log( )
log log2 4 23 3
21
1
0
1
2
x
x dx
... log
tan tan2 1
20
1
0
1
x xdx x x dx
2 .220671308254797711926... arctanarctan log
log2
12 24 4 2 3 3
2 32 2 2
2
20
x
x xdx
1 .2210113690345504761... 11
22
( )kk
.2211818066979397521... 1
1627 3 3 2 27 3 8 32( log ( ))
=1
3 231 k kk ( )
.2213229557371153254...
21
6
205
1445 1 2 5 2 5 ( ) ( ) ( , , ) ( , )
1 .2214027581601698339... e5
2 .22144146907918312350... 2 14
dx
x Marsden p. 231
= x dx
x
2
4 1
=d
1 20 sin
Marsden p. 259
= log cosh
cosh
x
xdx
xdx
x2 12
2
2
00
= log 11
2 40
xdx
.22148156265041945009...
122
2
1)(11
1
kkk kk
k
.22155673136318950341...
8
... 6 51 3
0
1
( )log( ) log
x x
xdx
... ( ) ( ) ( )( ) ( )3 11
42 22 2 i i
=x x
e edxx x
2 2
0 1
sin
( )
.22157... log ( )!
( )!
2
2
1
8
2 4
2
2
2
k k
kk
= ( )!!
( )! ( )
2 1
2 122
k
k kk
.221689395109267039... 1
13 1 3 1
13
2 16 3
2kk
k kk k k
( ( ) ) ( )
=1
31
1 3
2
5 3
2
1 3
2
5 3
2
i i i i
=
1
3
1
61 3
3 3
21 3
3 3
2( ) ( )i
ii
i
=
3
2
3 3 3
5 3
2
1
3
2
3 3 3
5 3
2
i
i
i
i
= 1
31 1 2 1 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /
.2217458866641557648... 1
63 4 2
2
41
(log log )( )
kk
k
.22181330818986560703... 2 2 26 2 1
22
20
log log( )
k
kkk
1 .2218797259653540443... log
kk
kk
112
2
.221976971159108564... 1
1142 kk
.22200030493349457641... ( )
!
/2 1 1 1
1
1
2
2
k
k
e
k kk
k
k
.22222222222222222222 =2
91
21 2
0
1
1
( ) arccoskk
k
kx xdx
=
1 19
3)(
kk
kk
...
1 )!1(k
k
k
B
... 1
6
2
3
2
3
1
21
21 2 1
1
log arctan ( )
k kk
k
H
2 .2226510919300050873... 3
4
1
2
1
2
2
0
e
e
k
kk
( )!
.2227305519627176711... 21 3
4 16
12
2 1
4
41
( )
( )
k
kk
= ( )( )( ) (
)
11 2
21
3
kk
k
k k k k
... G log2
2 .22289441688521111339...
8 4 1
1
2 21
2
4
=8
8
2 1
231
11k k
k
kk
k
( )
8 .22290568860880978... 2 1
1
2
1
k 1
k
k
k
k
k
e
k
( )
!
/
2 .2230562526551668354... 1
29 3 3
3
3 321
22
( log )( )
k k
k
kk
k
=1
1 230 ( )(k k )k
.2231435513142097558...
1
1
11 4
)1(
5
1
5
1Li0,1,
5
14log5log
kk
k
kk kk
= 21
2 1 92
1
9 42 10 0( )k
dx
ekk
x
arctanh1 K148
.2231443845875105813... H
kk
kk
( )
( )
3
1 3 1
.2232442754839327307... 2 1 1 21
2 1
1
2 1 2 2
1
1
1
1
sin cos( )
( )!( )
( )
( )!( )
k
k
k
k
k
k k k k
= x xd2
0
1
sin x
=log sin log
sin2
21
41
1x x
xdx
x
dx
x
e
.22334260293720331337... ( ) ( )
( )!
2 2
21
k k
k
1
k
.2234226145863756302... ( )
!
1
10
k
k k
.22360872607835041217... ( ( ) ) k
kk
1 2
2
98 .22381079275099866005... 267 1 9
0e
k
k
k
k
( )
!
.223867833455760676... 1
4 2
1 1
4 22
2 2
e e
csch
=( )
1
2
1
21
k
k k J125
.2238907791412356681... Jk
k
k
k
k
k
k0 0 1 2
0
1 2
21
2 1 11 1
( ) (; ; )( )
( !)
( )
( !)
F LY 6.115
= ( )( )!
( )( )!
11
2
21
2
2
0
12
0
k
k
k
kk
k
k
k
k
k
k
.22393085952708464047...
2
2
12223
)1log(1)12(
)1(
4
kkk k
k
k
k
kk
.22408379...
11 1
)(
)1(
)(
kk k
k
kk
k
3 .2241045595332964672... ( )( )
! /
11 3 13
2
2
1 32
k
kk
k
k k
k k
k k
e k
.224171427529236102395...
3 82 1
2 11
log( )
( )
k
k kk
... ( )( )
12 1 1 1
22
2 21
k
k k
k
k kkLi
k
.2244181877441891147... 7
361
7
6 71
( log )
k Hkk
k
.2244806244272477796... )1(22
2log
33
5
= ( )
logk
k kk k
kk k
1 1
3
1
3
1
21
1
1
2
2
... e
.2247448713915890491... 3
21
2 1
2 31
( )
( )!
k
k kk
!!
1 .2247448713915890491... 3
2
1
12
2
0
kk
k
k
= 11
6 30
( )k
k k
... 8 1
51e
ex
dx
x
sinh
1 .2247643131279778005... 1
18
3
6
1
92 21
cot
kk
.2247951016770520918... 1
2 4 2 2 2
1
8 1
2
821 1
cot
( )
k
k
kk
k
.22482218262595674557...
2 3 3
1
3 1
2 1
322 1
cot( )
k
k
kk
k
.224829516649410894... 1
8132 3 24 2 27 3
1
2 32
31
( log ( ))( )
k kk
.225000000000000000 = 40
9
.22507207156031203903... 1
0
3arctan2
2log
324dxx
.225079079039276517... 1
2
.225141424845734406... 22
2 11
2 20
1
G Edk
k
log ( )k GR 6.149.1
.2251915709941994337...
1
1
!)!1(
2)1(
k
k
k
.2253856693424239285... )()(14
1
16
)3(3333 iLiiLiLi
=
1
02
2
13
2
1
loglog
x
dxxx
xx
dxx
=x dx
e x
2
20 1
1 .225399673560564079... 2
2 1
1
20
e
e e kk
( )
1 .22541670246517764513... 3
4
1 .22548919991903600533...
2
2
1
1
coth(log )
... 1
184 2 2 2 3 2 2 1
21 2
1
arccot log log ( )k kk
k
kH
.2256037836084357503... H k
kk
2
1 8
1
74
8
72
4 2
4 2
log log
1 .2257047051284974095... ( )31
kk
.2257913526447274324... log ( ) log( )
21
2
222 1
1
k
kk
k
kk
k K ex. 207
.226032415032057488141... 8
1
6 51
arctan x
xdx
3 .22606699662600489292... ( ) ( ( ) )
1 21 3
1
k 1k
k
...
42
4
2 10
erfik k
k
k
!( )
.2265234857049204362... e
12
2 .2265625000000000000 = 285
128
1
54 0
1
5 54
4
1
, , Lik
kk
... 1
4
5
44
0
4
x e dxx
.226724920529277231...
8 3
.22679965131342898... 287
4
153
22
77
22
2 1 22
3
1
3
log log( )( )(
k
k k kk
k )k
H
.226801406646162522... 2
7
84 2 2
3
8 8cot log logsin logsin
=1
8 21 k kk
.2268159262423682842... 1
972 2 168 2 149
2 42
1
log log( )
H kk
k k
1 .2269368084163... root of ( )x 5
... 2
272 3
2
3 2
2
31
, ,log x
xdx
.22740742820168557... Ai(-2)
.22741127776021876... 3 4 21
2 2 3
1
10
1
2 21
log( )( )
( )
( )kk
k
kk k k k
= ( )!!
( )! ( )
2 1
2 1 21
k
k kk
1 .22741127776021876... 4 4 21
4 1
22
0
log( )k
k k
k
k
=1
121
22k k
k
k
kk
k( ½)( )
( )
=( )
( )
k
k k
!
!k
1
2
121
... 23
2
1 32
1
21
arcsinh
( )k k
kk
k
k
4 .2274535333762165408...
2
3 21 4
1 4
1
4log
( / )
( / )
1 .2274801249330395378... 3 5 66
372
12 22 1
1 1 12 42
12 3 4
( ) ( ) log
H
k k k kk
k
1 .22774325454718653743... H
k kk
k ! 21
1 .2278740270406456302... 1
11 k kk ! ( )
1 .227947177299515679... log( ) ( , , )2 21
21 0
... ( )( )
arctan
12 1
1 3 1
32 1
k
k k
k
kk
k
k
k
2 .2281692032865347008... 7
16 .2284953398496993218... e erfk
k
k
2
0
12
22
!!
= 8
35
1
1
2 2
2 4
p
p pp prime
...
2
)(
2 2 2
)(2
)1(
221
)2(
)(
k
k
k s ks
k
kkkk
k
.22881039760335375977... )1,2,2(1
)2,3(2
)5(11)3(
2 123
2
MHSjk
MHSjk
... dxxx
x
1
34
222 )1log(
48
7
24
2log3
.2289913985443047538... G
k
k
k4
1
4 2 20
( )
( )
=1
8 6
1
8 22 21 ( ) ( )k kk
=x dx
e e
x dx
x xx x2 20
3 11
log
=x x
xdx
log
1 41
.2290126440163087258... e
ee
H
e
kk
kk
11 1
1 1
1
(log( ) )( )
.22931201355922031619... H
kk
kk
(2)
51
... log
( ) ( )
2
21 1 1
x
x xdx
.229337572693540946...
1
21
1 1
21
loge ek
k k
.2293956465190967515... I e ee
k
k
k0 0 1 2
0
2 1( ) (; ; )( !)
F
... 3
22 1
11
2
log ( ) ( )k
k
k
kk
= log 11 1
12
k kk
... 1
8
1
2
1
4
1
2
1
2 1 4
1
1
cos sin( )
( )!
k
kk
k
k
.229637154538521829976... 3
22 1
11
2
log ( ) ( )k
k
k
kk
.22968134246639210945... 1
2 2
1
2
1
2 2
1
1
sin( )
( )!
k
kk
k
k
... sincos ( )
( )!2
1
1
1
2
1 1
2
1
2 2
k
k k GR 1.412.1
= sin1
21
3x
dx
x
2 .2299046338314288180... ( )
( )!cosh
2
2 2
1 1
12
1
k
k k kk k
.2299524263050534905... 24
25
3 2
5 10
1
4 51
log
( )
k kk
.2300377961276525287... 4
2 1
2 1 2 20
1
x x
x
arccos
( )
dx GR 4.512.8
.2300810595330690538...
1
4
5
21
5
2 1 21
cos( )
kk
.23017214130974243... 1
2 2
1
2 2
12
21 8
1
1eerfi
kk
k
k
kk/
( )
!
... 1626 1
2 30
e k k
k
kk
( )
! ( )
.2302585092994045684... log10
10
3 .23027781623234310862... 1
22
2
1
4 120
csch
( )k
k k
.23027799638651103535... 1
4 21k
k k( )
...
1
2 3102k j s
jks
.23032766854168419192... 5 5
121
1
2
21
1
( )k
k k
k
k
.23032943298089031951... 1
6
... 1
4
1
2
1
2 2
i i i
2
i
= cos
)
x
e edx
x x
10
...
log
cos2
2
1
1 2 20 x
xdx
x
.230769230769230769 = 3
13
5
0
sin x dx
ex
9 .2309553591249981798... k
kkk
4
1
.2310490601866484365... log ( ) ( )2
3
1
3 3
12
20
1
1
k
k
k
kkk k
kk
J292
= dx
x x
dx
x x
dx
e x41 0
203 1 3 2
( )( ) 3
1 .23106491129307991381... cosh cosh sinh sinh cosh sinh 2 2 2 2
= eii
.231085022619546563... ( )
!
k
k kk2
2
... log ( ) 2 11
kk
.2312995750804091368... k
kk
!!
( )
! 3
0
1 .23145031894093929237...
2
1122 2 1 2
1
2
log ( log )( )k H
kk
kk
.23153140126682770631... 1 1
1
1
1 e
k
k
k
kk
/
( )
!
.2316936064808334898...
2
1Ai
... k
kk
kLi
kk k
11 1
1 12
22
2
( ) log
2 .23180901896315164115... 1
21
2
( )kk
.2318630313168248976... 2 2 21
20
log( )
kk
k
1 .2318630313168248976... 2 2 2 121
log( )
k kk
k
2 .2318630313168248976... 2 2 2 21
20
log( ) k k
kk
1 .23192438217929238150... ( )( ) ( )
12 1
1 1
1
kk
k
k k
.2319647590645935870... 9
212
2 80
e k
k kk
!( )
.23225904734361889414... 6
2
91 2 2
0
1
K x x( ) arcsin( )dx
.2322682280657035591... 351 14 3 126 3
588
1
3 71
log
( )k kk
.2323905146006299... ( )log
( )
112
k
k
k
k k
1 .2329104505535085664...
1
0
2
1
log)3,3(16)1)3((16
x
dxx
3 .2329104505535085664... 16 3 11
2
0
1
( ( ) )log
x dx
x GR 4.512.8
= 16 3 181
2
2 20
( ) / /
x dx
e ex x
...
1
2)12(
51
2
5cos
4
1
k k
... 2 2 1
2
3
2
1
21 2
i ( / )
.2331107569002249049... ( )3
32
3
... 4 48
2
2
2
0
1
log
arctan logx x dx GR 4.593.1
.23333714989369979863...
( )2
41
kk
k
1 .2333471496549337876... Im ( )( ) 2 i
1 .2334031175112170571... arcsinh2
.2337005501361698274... 2
22 28
11
2 1
1 1
2
( )
( )( ( )
k
k k )
kk
k
=( )!( )!
( )!
k k
k
k
k
1 1 2
21
= dxx
xx
1
02
2
1
log
=log x
x xdx4 2
1
1 .2337005501361698274... 2 2 2
2 32 08
1
1 2 1
1
k k
k
k
k kk k
( )
( )
!
( )!!( ) J276
= ( )( )
( )( ( ) )2
3 2
4
1
2 12 12
0
2
1
kk k
k k
= 1
4 1
1
4 32 21 ( ) ( )k kk
= 2
221
k
k k
kk
=
1 1 2
)(
j kjk
jk
= log x dx
x20
1
1 GR 4.231.13
= log x dx
x40 1
= arctan x dx
x1 20
= arctan x dx
x
2
20 1
GR 4.538.1
= arcsin x dx
x1 20
= K kdk
k( )
10
1
GR 6.144
= E k GR 6.148.2 dk( )0
1
.2337719376128496402... ( )
!
( )
( )!!/k
k
k
ke
kkk k
k
k2 21
1
22 1
1 2
1
... sin1
31
2x
dx
x
.23405882321255763058... 1
2 2 2 2
1
21
1
21
1
2
cot
=k
k k
k k
kk
k
1
2
2 2
231 1
1 ( ) ( )
.2341491301348092065... 1
6 1 61
0
1k
kk
k
k
( )
.23416739426215481494... ( )3
6
1
36 616 3
1
k
k kkk k
.23437505960464477539... ( )
!
1
20
k
kk
.23448787651535460351... ( )2 1
2 1
11
1 2
k
kk
kk k
arctanh
.23460816051643033475... log k
k kk3
2
.23474219154782710282...
( )3
6
1
21
1
61
1
6
=1
36 6
2 1
65 31 1k k
k
kk
k
( )
.2348023134420334486... 1 11
40
1
21
Jk
k
kk
( )( )
( !)
.2348485056670728727...
4
12
4116
16 43
2
1
,( )kk
.23500181462286777683... 3 2
2
5
3 31
2log log
dx
x x
... 2 1
5 4 11
21
1
sin
cos( )
sin
kk
k
k
.235028714066989... 1
3612 3 2 3
3
2
3
22
sin csc sec
=( )
1
34 22
k
k k k
.2352941176470588 =4
17
4
0
sin x dx
ex
=1
2 41 2 1 2 2
0cosh(log )( ) ( ) log
k k
k
e J943
1 .2354882677465134772... 33
2 1
3 13
32 1
sech
k
k
k
kk k
exp( )
= exp logk
kk
3
32 1
=
13
)(
k k
kz
= 3 1 15 3
2
5 3
21 3 2 3 ( ) ( )/ /
i i
... 3 3
42 2
133
20
1 ( )log
log ( )
x
xdx
.23550021965155579081... 1
2 2 1
1
2 11
1
11
k kk
k
kk( )
( )
.2357588823428846432... 2 1
2
1
2
1
1
1
1
1
1 2
1
1e k k
k
k k k
k
k
k
k
k
k
( )
!( )
( )
( )!
( )
( )! !
.23584952830141509528... 89
40 CFG D1
.235900297686263453821...
5
15log5log2log42log4
2
1
4
12
222 LiLi
=H
k kk
kk
k
kk5
1
41
1
21
( )
.2359352947271664591... ( )( )!
/11
1
1
e e ek
k ee
kk
... 5
3 1 6 50
dx
x /
.23605508940591339349... 1 28
2 2
2 2 0
4
log log tanlog
sin log(sin )/
x x dx
= 1 2 2 12 2
2 2
log ( )
log
.236067977499789696... 5 21
3
1 .236067977499789696... 5 11
2 1
21
1
( )k
k k
k
k
=
( )
( ) !
3 1
2 20
k
k k kk
8
Berndt Ch. 3, Eq. 15.6
2 .236067977499789696... 5 2 1 21
5
23
0
kk
k
k
4 .236067977499789696... 3 2 1
... 12 2 9 3 3 36
1
6
2
3 21
log log
k kk
2 .236204051641727403...
( ) ( )2 5
22
224
277 3
1 .2363225572368495083... 1
4 2 6
2
3
1
3 221
cotkk
.23645341864792755189... 3 2 1 2 11
2 1
1
1
cos sin( )
( )!( )
k
k k k
1 .2365409530250961101... 3
4 2 2
2
0
2
0
e k
k
k
kkk k
! ( )!!
... csc ( )3
6
1
18
1
9
1
2 20
k
k k
.2368775568501150593... 1
216
5
6
1
32 2
2
20
( ) ( )
x xdx
e ex x
=log2
31 1
x
xdx
.2369260389502474311... 25
64
1
2
1 6
0e
k
k
k
kk
( )
!
... 9 3
4 1
2 3
31
3
30
1 ( ) log
( )
log
( )
x
x
x x
x 1
.2374007861516191461... log( )3 3
.237462993461563285898... 2
4
3
1
5 20
( )
/
k
k k
=
2/
02
2
)sin1(
sin
dxx
x
... 252
15log
4
3
24 22
2
Berndt Ch. 9
9 .2376043070340122321... 16 3
3 CFG D11
17 .2376296213703045782... k
kk
3
1 1( ) !!
.23765348003631195396... 2
2130
137
1500
1
5
k kk ( )
.2376943865253026097... ( ( ) ) jkkj
3 111
98 .23785717469155101614... 24 212 1
22 3
30
log( )
e x dx
e
x
x GR 3.455.2
.2378794283541037605... 1
144
1
6
2
31 1
3 31
( ) ( ) log
x dx
x x
... ( ) ( ) ( )
( )
( )2 3 2 2
1
1 21
k
k kk
.23799610019862130199... 113
2 3k ks d
k log( )
s
.2380351360576801492... ee
erfk
k
k
2
1
2
1
0
( )
!!
.238270508910354881... 8 21
2
146
15
1
2 2 70
arctanh
kk k( )
6 .238324625039507785... 9 2log
.2384058440442351119... 2
12e
16587804425989... = 2 2 2 3
20
2 3 2
16 2 2 1
( log ) log
x dx
x
1 .2386215874332069455...
1
/1
1
1 2
1!
)2()1(
k
k
k
k ek
k
.2386897959988965136... ( )
1
120
k
k k k
.23873241463784300365... 3
4
.2388700094983357625... 1
16
3
2 1
3
1
( )( )
ee
k
k !k
2 .2389074401461054137... k H k
kk
( )3
1 2
2 .23898465830296421173... ( ) ( )3 5
.2391336269283829281... 2 1 11
2 2 31
2 2
200
cos sin( )
( )!( )( )
( )
( )!
k
k
kk k k
k
k
=log coslog2
1
x x
xdx
e
.2392912109915616173... Ei ekek( ) log( )
/ 1 1 1
1
.23930966947430038807... ( )k
kk
3
32
= 1
1
1 13
26 3
212 3
2k k k kk k k
k
= ( ) ( ) ( )3 1 9 3 1 9 3 11 1 1
k k kk k k
.2394143885900714409... 1
8
2
4
1
42 21
cot
kk
2 .2394673388969121878... 2 3 2 4 ( ) ( )
... 1 2
3 6 41 3
43 3
logcot ( cot (
= ( )
12
2
k
k k k
.2396049490072432625... log( )
e
e2 1 J157
... 12 3
3
1
2
1 22
2 1
1
1
arcsinh( )
( )
k k
k k
kk
.2396880802440039427... 1
831 kk
.2397127693021015001... 1
2
1
2
1
4 2 1
1
1
sin( )
( )
!
k
kk k
...
1 )(
1
k k k
.2397469173053871842...
( )log
32
31
k
kk
.2397554029243698487... 21
2 2
1
2 22
1
1
sin( )
( )!
k
kk k
.23976404107550535142... 1
22 2
1
23 2
1 2
22 0 10
Jk k
k k
k
( ) (; ; )( )
!( )!
F
.2398117420005647259... GR 4.381.1 ci x x dx( ) log sin10
1
= ( )
( !)
1
2 21k k k
k
.239888629449880576494...
1 2
1)2()2(12log
4
7
k k
k
=
kk
kk
42
2
2
11 1
2log
... dxx
x
1
06
22
)1(
log
6
2log5
303
1
.2399635244956309553... 1
2
1
4,
.24000000000000000000 =6
25
1
61 0
61
( , , )k
kk
.24022650695910071233...
11
1
12
)1(21)1(2log
2
1
kk
k
k
kk
k
H
k
H
= dxx
x
1
0 1
)1log( GR 4.791.6
=
0
2
1
1
0 1
loglog
1
)1log(xe
xdx
x
xdx
x
x
.2402290139165550493... 2 1 2 2 11
10
1
log( ) log e
e
edx
x
x
.24025106721199280702... 3
25
.2404113806319188571... ( )3
5
.2404882110038654635... ( )( )
13
13
1
k k
kk
H
1 .2404900146990321114... 1
1 3
1
2 3/ /
.24060591252980172375... 1
2
1
2
1
4 2 1
22 1
0
arcsinh
( )
( )
k
kk k
k
k
.2408202605290378657... 1 2 2 32 3
1
( ) ( )( )
k
kk
= ( ) ( ( ) )
1 11 2
1
k
k
k k 1 Berndt 5.8.5
.2408545424716093105...
12
21
8k k
k
... 1
1 21 ( )! (k kk
) Titchmarsh 14.32.1
... 2 24
23 3
16
22
0
1
log( )
arctan logx x dx
... 1
6
2
3
1
1 2
1 12
1
( ) ( )!
( )!
k
kk
k
k
13 .2412927053104041794... 257
32 2
5
1
e k
k kk
!
... 2 2 3
2112
2
2
2
6
3
8 2
log log ( ) ( )k
kkk
.2414339756999316368... 5 2
4
5
8 2 22
log
( )
1
k
kkk
.24145300700522385466... 1
1
.241564475270490445... log( )
log
4 1
1
2
20
1
x
x
dx
x GR 4.267.2
=x x x
x xx dx
log
loglog( )
112
0
1
GR 4.314.3
.24160256000655360000...
( )
( )
2
5 1
1
512
1
kk k
k k
... 15 33
6
Associated with a sailing curve. Mel1 p.153
...
2
20
1
1
( )
( )sin/
e
ee xx dx
))
.24203742082351294482... ( (22
kk
1 .2420620948124149458... 1
2 1
2
2 11 2
21
1
11( )log log( )
( )
`k
k
k
kk
kk
kkk
k
=
1
1
2
)(
kk k
k
.242080571455143661373... logsin/3
32
27 8 8
50
x
xdx
.24214918728729944461... 112 9 24 2
271
2
0
1
loglog( ) logx x x dx
.242156981956213712584... ( )( )
14 1
1
1
kk
k
k
.2422365365648423451... 3 2
2 20
2
3 11128
x dx
e e
x dx
x xx x
log
.24230006367431704... ( )
1
230
k
k k
.242345452227360949... c231
63 2 3 ( ( ) ( ) ) Patterson Ex. A4.2
.24241455349784382316... ( )
!
1 1
1
k ek
k
H
k
3 .24260941092524821060... 1
2 (log )k kk
.24264068711928514641... 3 2 48 1
2
1
k
k
k
kkk ( )
4 .24264068711928514641... 18 3 2
...
1
11
kk k
k
k
kLi
2 .24286466001463290864... 8 13 38 3 3
12
4 3 2
3 31
1
( )( )
k k k k
k kk
= ( ) ( ( ) ( ) )
1 23
1
k
k
k k k
.24288389527403994851... 2 3 3( ) ( )
...
1
21
21
)!12(
)!()!(
23 k k
kk
= 1
0
32
2
)arctan(1
dxxx
x
.24314542372036488714... 1126
9 7 5
2 2
9
1
2 921
log
( )k kk
2 .2432472337755517756... 128
1
48
7
120
24
= k kk k k
kk k
3
1
2 4 2
2 42
2 11
1( ( ) )
( )
( )
.2432798195308605298... e
e
k
kk
k
2
1
12
0( )
( )
!
B
.24346227069171501162... ( ) ( )
!/
/k k
ke
e
k kk
kk
k
1 11
2
11
21
.24360635350064073424... 5 82 21
ek
k kkk
! ( )
.2436903475949711355... 9G
.2437208648653150558... 83
23
1
21 3
1
2 30
log ( , , )( )
( )
k
kk k
.2437477471996805242...
4 2
1 2
2 2
1
4 3 104
2
40
1
1
log( ) ( )k
k k
dx
x
x dx
x 1
=dx
e ex x30
.243831369344327582636... 7
2
1
2
3
2
3
2
3
22 1
1
cot ( )k
k
k
.24393229000971240854...
k kk kk k
( ( ) ) 3 11
113 3 2
2
.2440377410938141863... ( )1
.2441332351736490543... 1
4
1
4
22
1
42 2
1
21
e e k
k
k
csch
2
( )
=sin2
10
x dx
ex
.24413606414846882031... log9
9 J137
.24433865716447323985... k
kk
1
2
21 2( )
... 2 2 22 1
2 21
1
log( )!!
( )! ( )
k
k k kk
.24483487621925456777... 41
4
1
4 2 12
1
sin( )
kk k k
.2449186624037091293... e
e
1
1 J148
.24497866312686415417... arctan( )
( )
1
4
1
4 2 12 10
k
kk k
6 .24499799839839820585... 39
.2450881595266180682... 63
2
3 3
22 2
1
6
1
6 21
loglog hg
k kk
= ( )( )
11
61
1
kk
k
k
= dxx 1
0
6 )1log(
.24528120346676643... 2 2
20
4
16 42 log
cos
/
Gx dx
x GR 3.857.3
= arctan 2
0
1
x dx
.2454369260617025968... 5
643
0
1 x xarcsin dx
.24572594114998849102...
22
2
1
sin11log21log2)1(
8 k
iiii k
keee
e
i
.245730939615517326... 35
2 .245762562305203038...
0
2
2 )1()!2(
42sinh
2
2cosh
2
1
44
3
2
1
k
k
kk
e
e
.2458370070002374305... 1
2
1 1
2 0
1
sin cos sin
e
x dx
ex
=sin log x
xdx
e
21
... dx
e xx ( log11
)
.2458855407471566264... ( ) log2 2 16 12 21
4 3 21
k kk
= ( )( )
12
41
1
kk
k
k
2 .2459952948352307... root of ( ) ( )1 1
2x
.2460937497671693563...
( )2
221
kk
k
...
4
3
4
12log)2log23(log2
6 22
2
LiLi
8 .2462112512353210996... 68 2 17
.2462727887607290644...
( , , )( )1
44 0
1
4
1
41
k
kk k
.24644094118152729128... 1
14 3 21 k k k kk
1 .24645048028046102679... 2 1 2 2 1 2log( ) arcsinh arctanh 1
2
=1
2
1 2
2 1log
= 1
2 2 1 20 1k
k
k
kkk
H
( )
O
= 1 12
2 31
( )( )!!
( )k
k
k
k !! J141
=( ) ( )!!
( )!!
1 2
2 1
1
1
k
k
k
k k
=dx
x x( )
1 1 2
0
=
1
0 1)1( xx
dx
=
2/
0 sincos
xx
dx
.24656042443351259912...
1
22
2
1
2
)18(
8
8
)2(
22csc
2sin2
32 kkk k
kkk
.2465775113946629846...
( ) ( )2
2 1
2 1
4 111 1
k kk
kk
k
.2466029308397481445... 3
2
3
2 32
1
log( )
1
H
kk
kk
... 1
1728
2
3
1
62 2
2
3 31
( ) ( ) log
x
x xdx
1 .2468083128715153704... e e ee kk
k
log( )( )
11
10
1 .2468502198629158993... arcsec
.2469158097729950883... 1
2 2
1 1e
dx
ekk
x
... dzdydxxyz
xyz
1
0
1
0
1
0
4
1
log
30
.2470062502950185373... log ( )3
2 6 3
1
9 3 321 2
3 21
k k
k dx
x xkk
k x
3 .2472180759044068717... 3 46
4096 30 3870720
14 6
61
( )
( )
( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
.24734587314487935615... 1
44 4
0
1
( log ) log ( ) sin x x dx GR 6.443.1
.24739755275181306469... 1
232 kk
.2474039592545229296... sin( )
( )!
( )
( )!!
1
4
1
2 1 4
1
4 2 2 42 10 0
k
kk
k
kkk k
.2474724535468611637... 7
4
4
3 23 2
log
.24777401584773147... pq k
kk
( )
2 10
... 33
2
1 12
4 12 21
Gk
k
k
k
( ) ( )
( )
.24792943928640702505... ( ) ( )2 12
k kk
.24803846578347406657... ( )
logk
k kk
k jkk
1
1
1
2
1
42
1
21
12
22
.24805021344239856140... 3
125
.24813666619074161415...
2 6 6
1
2coth
.24832930767207351026...
1
21
21
2
1
4 31
i i
k kk
= ( )( )
Re( )
( )
12 1
4
1
21
11
kk k
kk
k k
i
... 84
1
1 2
3 1
31
( )
( /
k
k k )
.24873117470336123904... 2
2 2
1 2 2 120
1
log arcsin
x
xx dx GR 4.521.3
.2488053033594882285... ( ) ( ) ( ) ( )( )
2 3 4 3 3 51
3
51
k
kk
.24913159214626539868... 7
360 6
4 2 4
40
x
xdx
cosh
1 .249367050523975265... sinh3
... log x
edxx
11
.2494607332457750217... 1
4 11
3
231
12
1k
k
k
kk
k
( )( )
.2498263975004615315... sin sinh( )
( )!
1
2
1
2
1
4 2 2
1
2 11
k
kk k
GR 1.413.1
.250000000000000000000 = 1
4
1
3 6
1
4 421
21
k k k kk k
= 1
1 12 ( ) ( )k k kk
J268, K153
= 1
2 132 1k k
kk k
( ( ) )
= k
kk 4 141
=
3
1
222
)1)()(1()1()1(
12
k
k
k
kkkk
k
= ( ) ( )
( )( )
( ) ( )
1
1
1
1 3
1
2
1
222 0
1
21
1
23
k
k
k
k
k
k
k
kk k k k k k k
= ( )( )
14 1
4 414
1
kk 1k k
k k
k
= ( ) ( ) ( )
1
3 4 1
2
4 1
1
1 1 1
k
kk
kk
k
kk
k k
= dx
x
x dx
x
x dx
x
x dx
x51
3
50
31
2
311
( )
log log
= x x xarcsin arccos0
1
dx
x
x
= x x dlog0
1
= log(sin )sin cos logsin sin cos/ /
x x x dx x x x d0
22
0
2
= log 11
13
x
dx
x
=
44
7
0
3
xx e
dxx
e
dxx
1 .250000000000000000000 = 5
4 22 H ( )
2 .250000000000000000000 = 9
4 3 11
kk
k
68 .25000000000000000000 =
1
5
3)0,5,
3
1(
4
273
kk
k
.25088497033571728323... ( )k
kk
1
2 12
... 12primesmallest;2,lglim 1)(
npnn
n
nppp
Bertrand’s number b such that all floor(2^2^…^b) are prime.
... 1
22 2
1
2 20 00
I Jk kk
( ) ( )( )!( )!
.2517516771329061417... 1
41111
3
2
3
22 2
1
16
1
2
3
21
HypPFQ , , , ,{ , , , },(( )!)
(( )!)
k
kk
2 .2517525890667211077... ( )10
.2518661728632815153...
4 5
5
2
1
10
1
4 521
coth
kk
.251997590741547646964... 3
2
3
3
1
41
1
1
3
( ) ( ) ( )
( )
k
k kk
6 .25218484381226192605... 7 3 4 22
2 2 1
2 1
2
31
( ) log( )
( )
k
k kk
=k k
kk
2
1
2
2
( )
.252235065786835203... 11
2 11
10
1
e e
dx
e ex xlog log( )
.2523132522020160048... 1
5
4 .2523508795026238253... Ik
k
k0 0 1 2
0
2 2 1 22
( ) (; ; )( !)
F
.252504033125214762308...
sec hi i
2
1
2
1
2
...
( )( )
36
2
21
k
kk
.2526802514207865349... arccsc4
1 .2527629684953679957... Li15
7
.25311355311355311355 =691
2730 6 B
... 2 8 3 ( )
2 .25317537822610367757... k
kk
!!
( )!!21
.2531758671875729746... = 11
2
1
2 1
1
1
erf
k k
k
k
( ) ( )
!( )
.253207692986502289614...
7
1 .25322219586794307564... 1
2 3 31
( )
( )f k
kk
Titchmarsh 1.2.15
.2532310965879049271... 9 3 9 3
8 12
1
3 2
2
3 21
log
k kk
... 2/
0
223
cos848
dxxx
1 .25331413731550025121... 2
i i ilog
... 1
42 2
2 21 10
I Jk
k kk
( ) ( )( )!( )!
1 .25349875569995347164... 1
12 kk !
...
0
2
2
3
)3log(
32
12log
x
x
.25358069953485240581... 1
30 ( )!k k!k
.25363533035541760758... 2
32
5
2
1
2
1
21 1
1
120
Fk
k
k
kk
, ,( )
( )!
... 24 81 4
0
1
e e dx /
x
.25375156554939945296... dx
x x
dx
e ex x4 11
3 20
.2537816629587442171... 65
6 453 3 5
1
1
2 4
5 21
( ) ( )( )k kk
2 .2539064884185793236... 2
2
2
21
k
k
k
.25391006493009715683... 1
42
1k
k
.2539169614342191961...
12 2
1
2
11
1log
2
1
kk
kk
Likk
k
.25392774317526456667... 1
2 21 ( )k k
k
.2541161907463435341... 1
44 0
1
4
1
44 41
, ,
Likk
k
.254162992999762569536...
1
sindx
e
xx
.2543330950302498178... 2
4 2 6 CFG D1
.25435288196373948719...
6 3
2
3
3
6 131
2
log log dx
x
.25455829718791131122...
011 )72(!
1,2
11,
2
9F
9
1
k kk
k
.25457360529167341532... 1
32 4 2 7 214 8 2 8 2 7 2 22
( )sinh sinh
i
i i
csch
=1
2 1 21 ( )k kk
.2545892393290283910... 1
2 21
4 4
2 2 4
0
cosh( )!
/ /
16
e e
k
k
kk
1 .2545892393290283910... 1
2 21
1
2 1 4 1621
4
1
cosh( ) ( )!
k kk
k
kk
= sin sinh/
x x d0
2
x
23 .2547075102248651316... 3
4
3
11 .2551974569368714024... 4 13
320
k
k k
k
2 .25525193041276157045... 21
2cosh
.25541281188299534160... arctanh1
4
1
4 2 12 10
kk k( )
AS 4.5.64, J941
4 .25548971297818640064... 7
2
422 ( )
.25555555555555555555 = 23
90
1
2 1 2 70
( )(k k )k
K133
.25567284722879676889... 1
2515 8 5
1
2
1 1
2 1
1
1
( )( ) !( )!
( )!
arcsinhk
k
k k
k
1 .25589486222017979491... 1
20 ( )!k k!k
.25611766843180047273... ( )4 15
6
.2561953953354862697... 4 2 212
622
2
0
1
log log arcsin log
x xdx
.2562113149146646868...
1
8 2 2 2 22 2
2csch sech sinh
=( )
1
2 1
1
21
k
k k
.2564289918675422335... 7
33
1 3
1 3 2
1
e k
k kk
/
( )!
.256510514629433845... log( )
( )
1
1
2
0
1
x
x xdx
60 .25661076956300322279... 333
0
ek
k
k
k
!
1 .2566370614359172954... 2
5
.2566552446666378985... H k
kk
( )3
1 5
2 .2567583341910251478... 4
... csc2
... 1
82 2
2 20 0
2
0
I Jk
k kk
( ) ( )( )!( )!
1 .2570142442930860605... HypPFQ { },{ , },
( )!
11
2
1
2
1
16
1
220
k
k
kk
2 .25702155562579161794... 16 253
6 23
1
log
Hkk
k
1 .25727411566918505938... 5
.2574778574166709171...
2 2 21
2 12
1
log log( )
( )
k
kkk
.25766368334716467709... dxxx
xLi
1
0
2
22
)2)(2(
log
2
1
4
1
8
2log
48
.25769993632568292... ( )
1
2
1
31
k
k k
.25774688694436963...
1
)1)1((1k
k kH
1 .25774688694436963...
( )log log
kk
k k
k
kk km
km22
2 21
=1 1
11
1 11 2 1k
k
k
k
kH k
k
k
kk
k
log ( )
( )( ( ) )
.2577666343834108922... H
kk
kk
( )
( )
3
1 2 2 1
.2578491922439319876... 11 1
3
22 2
1
2
20 1 1 1
0eI I
k
k
k
k
k
kk
( ) ( ) ( , , )( )
!
F
...
1
125
2
52
2
2 30
( ) x dx
e ex x
.25794569478485536661... ( ) ( )
( )4 2 3
2
35
1
1
2
4 21
k kk
... 2 21 1
0
sinh /( )! 2 1
k
k
404 .25826782999151312012... 1
4
2 2
0
e ek k
ke e
k
sinh cosh
!
.25839365571170619449...
( )
( )
k
kk
1 1
2
.25846139579657330529... = Likk
k3 3
1
1
4
1
43 0
1
4
( , , )
=log( / ) log1 4
0
1
x x
xdx
.2586397057283358176... 2
12
226
2 21
2
1
2 1
log( ) ( )
( )
k k k
k kkk k
.258742781782167056... 3
42
7
12 1
2
0
log
xe dx
e
x
x GR 3.42.0
3 .2587558259772099036...
22
2
2
062 2 2 2
log loglog
xdx
ex
.25881904510252076235... sin ( )
12
2
43 1
6 2
4
AS 4.3.46, CFG D1
= 2 3
2
CFG C1
.25916049726579360505... 2 2 21
2
1
2 2 30
arctan( )
( )
k
kk k
.259259259259259259259 =7
27
1
72 0
7
2
1
( , , )k
kk
...
1 21
k
k
kk
1 .2599210498948731647... 2 11
3 23
0
( )k
k k
.2599301927099794910... log8
8 J137
.2602019392137596558... ( )( )
11
21
1
2
1
2 11
2
12
1 1 2
3k
kk k
k k
kk
k
... ( )2 11
kk
.26022951451104364006...
ie e e
k k
ke e
k
i i
4
1
22
2 2 2
1
cos sin(sin )sin cos
!
.26039505099275673875...
1 2!
)1(12logk
kkk
kk
Be Berndt 5.8.5
.260442806300988445...
2
23
41
4
1
10
1
2log loglog
x
dx
x GR 4.325.4
=log x dx
e ex x
0
.2605008067101075324... 11
51
kk
.26051672044433666221... ( ) ( ( ) )
1 1 3
2
k
k
k
.26054765274687368081... 1
2
1
2
1
2 1 41
sinh( )!
k k
k
1 .260591836521356119... cosh( )!
1
2
1
2 20
k kk
.2606009559118338926... 2
2 2112 3 4 3 3
1
2
1
3 1csc cot
( )
kk
J840
.2606482340867767481... 81 3
24
3
2
9 3
281
1
3
2
5 41
log( )
k kk
=( ) ( )
1 4
3
1
1
k
kk
k
.26069875393561620639... ( )k
kk 50
...
0 1
2
),2(!
),2(
k kkSk
kkS
.2609764038171073234... 2
31
3 3
24
2 1
2
( )
( )
k
kk
.26116848088744543358...
21
21
2 1
21
22
1
log(cos ) ( )( )
( )!k
kk
k
B
k k AS 4.3.72
... 21
2 2
1 1
2 22
1
arcsin( )!( )
( )!
k k
k kk
!
1 .26136274645318147699... 1
3 21 k k kk
.26145030431082465654... ( )3 33
21
31
3
i i
4 .2614996683698700833... 742
343 31 3
5
1
ek
k kk
/
!
... arctanh1
2
3
4
1
4 2 11
log( )k
k k k
1 .261758473394486609961...
eG
1 .2617986109848068386... 11
2 2 11
kk k( )
= arccosh x
xdx
dx
e ex41
30
x3
.26179938779914943654... 12
1
6 3
1
4 2 1
2
02 1
0
( )
( )
k
kk
kk k
k
k
=x dx
x
5
120 1
=
07
333 cossindx
x
xxx
... 11 2
122
2
2
2
6
7 3
8
2 2
( log )
loglog log ( )3
= 1 21
2
1
2
1
2 12 3 31
log( )
Li Lik kk
k
1 .26185950714291487420... log3 4
.26199914181620333205... ( )4 3
4 11
kk
k
.2623178579609159738... dxx
xx
1
0
22
1
log
13500
256103)3(16
1 .2624343094110320122... e
e iii
ik
k
/
/
2
201
1
1
.2624491973164381282... | ( )|( ) k k
kk
1
3
1
1
.2625896447527351375...
( ) ( )k kk
k
0
1 2
1 .2626272556789116834... arctan
.2627243708971271954... log
kk
kk
12
2
.262745097806385042621... ( )k
k
k 2 121
.26277559179844674016... 2
228
25
18
4 1
2 2 1
log( )
k
k kk
= ( )( )
11
22
kk
k
kk
...
2 !!
11
k k
... 7 3
32
2
2 20
( )
t dt
e et t
.2631123379580001512... 1
42 2 2 0 1 2 0 1 i i i( , , ) ( , , )
=( )
!( )
1
2
1
21
k
k k k
.263157894736842105 =5
19
...
2
75
1
1
2 2
2 2
p
pp prime
.2632337919139127016... ( )( )
!!
11
2
k
k
k
k
.26360014128128492209... 1
62 2
5
2
1
4 22 1
2 11
Fk
k
kk
k
, , ,
( )
.26375471929963096576... H
kLi
k
kk
kk
( )
( )
2
12
025
1
4
1
5 1
5
4
1
5
.26378417660568080153... 2
3 2118
4
92
16
27
1
2 3
logk kk
5 .263789013914324596712... 8
15
25
, the volume of the unit sphere inR
.263820220500669832561...
1
2
2 )!1(
)1(1
6)(
k
kk
kk
BieLi
[Ramanujan] Berndt Ch. 9
.2639435073548419286... 2
2 21
2
1
21
log( )k
k k k
= ( )!
( )!( )
k
k kk
1
2
121 4 1
=2
02
1
0
2 11log)1log(
x
dx
xdxx
= GR 4.591 arcsin logx x d0
1
x
2 .2639435073548419286... 2
2 log
= log log( )
11
12
220
1
0
xdx
x xdx
...
2 2
)()1(
kk
k kk
1 .2641811503891615965... k
kk ( !)31
.26424111765711535681... 12 1
2
1
1
1
10
1
1
4
0
e k k
k
k
k
k
k
k
k
k
k
k
( )
!( )
( )
( )!
( )
( )!
=1
2 1
1
11 0( )!( )
( )
( )!k k k k!k
k
k
= ( )! !
( )log
!
11
12
1 1 2
k k
k n
kn
k
k H
k n
k
k
= xe dxx
xdxx
e
log2
10
1
.264247851441806613... ( ) log ( )( )
3 116
3 1
4
3 1
21
2 3
4 1
2 1
0
kk
k k
.264261279935529928...
2
25
7
5 1
2
5 20
csc
( )
x
x
1 .2644997803484442092... 1
2 1
1
2 10 1k
k
k
kk
k
( ) ( )
.2645364561314071182... 9121
5
1
1 60
ek k kk
!( )( )
... 3
8 2
1
4 2 1
21 2
1
( )
( )
k
kk
k
k
k
k
4 .26531129233226693864... ( )k
Fkk
12
...
1 4
sin
1cos817
1sin4
kk
k
1 .26551212348464539649... log log21
2
4744 .2655508035287003564... 8
2
1 .2656250596046447754... 1
20k
k!
.2656288146972656805...
( )
( )
3
4 1
1
413 3
1
kk
k k
k
1 .2656974916168336867... 1
21 4 4 1 2 2 1 12 2
ee e e e si c( ) cosh ( )( ( ) ( i ))
2
( ) log ( ) ( ) ( ) ( )1 2 1 2 12 2 2e e ci e si =
H
kk
k ( )2 11 !
.26580222883407969212... sech21
2
22 2 cosh e e
.265827997639478656858... F kkk
( )2 1 12
.2658649582793069827... 2
3 122 3
G
log( ) Berndt 9.18
.2658700952308663684... 1
2 11
k kk
( )
4 .2660590852787117341... 1 1
11e e
ek
e
e
kk
( , , )
... Io ( )1
=1 21
22
0e
k
k
k
k( )!
( !)
= e dxcos
0
1
x
... 9
4
1
4
5
6
1
3
1
2 31 1
21
( ) ( ) ( )
( /
k
k k )
118 .2661309556922124918... 31
252
465 6
4 1
65
0
1
( )
(log )xdx
x GR 4.264.1
x dx
ex
5
0 1
.2662553420414154886... cos( ) 2
1 .2663099938221493948... ( )
1 2
1
k
kk
k
F
5 .26636677634645847824... 2
2
( )kk
2 .26653450769984883507... dx
x( )1
... 1
2 13 2
2
0
cothsin
cschx x
e ex x
.26666285196940009798... | ( )| 2
41
kk
k
1 .26677747056299951115... 8 2 2 2 21 2 1
0
log log( )
(k +1)8k
k
k
k
.26693825063518343798... dxex
x
e
e
1
02)1(
log
1
)1log(1
... 11
8 2
3
1
e k
k kk
!
.2673999983697851853... 1
22 2log
= log cos sin log 8 8
1
22 2 2 2
=1
41 k
k
k
cos
=
1
4 41 2 1 1 1 11 4 1 4 3 4i
( ) log( ( ) ) log( ( ) )/ / /
...
4
32log2log23log2
64
12
2
2 LiLi
=( ) log( / )
1
3
1 41
1 0
1kk
kk
H
k
x
xdx
5 .267778605597073091897... 2 2 7 31
22
0
log ( )sin
cos
x x d
x
x
... ( )
( )! (
1 2
1 21
k k
k k k ) Titchmarsh 14.32.1
.26794919243112270647... 2 31
6 1
2
1
kk k
k
k( )
.26794919243112270647...
k
k
kkk
2
)1(6
133
0
4 .2681148649088081629... 160
27
8
27
64
272
35 2
22
0
log ( )/x Li x dx
.268253843893107859...
6 4 23 3
21216 12 2
1 2 1
( ( ) )( )f k
kk
Titchmarsh 1.2.14
.268941421369995120749... 1
1
1 1
1e e
k
kk
( )
.26903975345638420957... erfi
ee x xx1
4
2
0
sin cos dx
... ( )
11
k
kk
k
F
.2696105027080089818... ( ) cos cos(log )
1
1 1 1
1
21 0 0
1k
kx
k
k
x
edx
x
xdx
=1
41
21
2
1
2 2
1
2 2
i i i i
=
2
1
2
1
224
1 iiii
4 .2698671113367835... e e x xdx
xx
cos sin(sin )0
GR 3.973.4
.270034797849637221...
2 2 22 20e
x dx
x
cos
.27008820585226910892... 16
105
.27015115293406985870... cos ( )
( )!
( )
( )!( )
1
2
1
2 2
1
2 1 2 10 1
k
k
k
kk
k
k k
.27020975013529207772... 8
1
2
1
4 141
2
arctan
x dx
x
.27027668478811225897... 2
2116
2
2 1
log log
( ) ( )
x
x xdx
1
.270310072072109588... 2
3
3
21 1
1
log ( )
k k
k
H
2k J 137
.2703628454614781700... log( )
( ( ) )2 11
12
k
k kk
=1 1
1 12 k kk
log/
K 124h
1 .2703628454614781700... log 2
=1
1 20
xx
dx
xcos
.2704365764717983238... 14 2 2
1
4 421
1
tan
k kk
4 .27050983124842272307... 5
23 5 5
1
5
2 2
0
( )
kk
k
k
... dxx
x
1
061
1log
32
32log
2
32log5
.270580808427784548...
1
3
4
)1,3()1(4
)4(
360 k
k MHSk
H
.270670566473225383788...
1
3
1
2
1
1
2 !
2)1(
!
2)1(
)!1(
2)1(2
k
kk
k
kk
k
kk
k
k
k
k
ke
3 .27072717208067888495... 2 212
2
3
3 3
2
3 3 4
40
2
loglog ( )
sin
/
x
xdx
1 .27074704126839914207... ( )
!
1
20
kk
kk
B
k
...
1
2
4
5,1
24
2 kk
kk
G
.27101495139941834789... cosh
1
2
1
2i i
=
1
21 )12(2)1(k
kk k
.27103467023440152719... ( )3 1
6 2 3 2 3 1 1 1
4
120
6
120
x
xdx
x
xdx
.2711198115330838692... ( )
1
2
1
31
k
k k k
... e
e
k
k kk16
1
16
1
4
1
2 2 11
sinh( )! 4
1 .27128710490414662707... 1
4 1
1
4
5
41
01k kF
k !( ), ,
1
.271360410318151750467... 1
2 2
1
2 2 21
sinh( )!
k
k kk
... cosh
sinh1 1
2
12
13
x
dx
x
... 212
2 2 22
2 2
0
2
log log logsin sin/
x x dx
1 .2717234563121371107... 0 10
21
22 2
1
2 1F1 ; ; ( )
!( )!
Ik kk
k
.2717962498229888776... 63 5 3 45 3
150
1
3 51
log
( )k kk
.2718281828549045235... e
10
1 .27201964595140689642...
.27202905498213316295... sinh
11
11
2
121
22k kk k
=
( ) ( ) ( ) ( )i ii i
i i
3 3
101 1
= )2()2(10)2()2(2
1iiii
1 .2721655269759086776... 6
9
.27219826128795026631...
2
1
2
1
28
2log22
iLi
iLi
iG
= dxx
x
1
021
)1log( GR 4.291.8
= log( )x
xdx
1
1 21
GR 4.291.11
=
log x dx
x1 40
1
GR 4.243
=x dx
e ex x 2 20
2log
GR 3.418.3
=
log(sin )sin
sin
/
xx dx
x1 20
2
GR 4.386.1
= log1
1 20
1
x
x
dx
x GR 4.297.3
=x dx
x x(cos sin )cos
/
0
4
x
= x x xarccos log0
1
dx
=arctan x
xdx
10
1
= log( GR 4.227.9 tan )/
10
4
x dx
= log((cot GR 4.227.14 ) )/
x d 10
4
x
.272204279827698971... 1
1242 kk
.2725887222397812377... 4 25
2
1
2 21
log( )
kk k
6 .2726176910987667721... 8 21
4
12 2 1
161
cotkk
.27270887912132973503... 3 2 21
2 11
2
e Eik
k kkk
log ( )( )!
.27272727272727272727 = 3
11
.27292258601186271514...
( )( )( )
12
1
11
kk
1 .27312531205531787229...
1
22
24
2
1)2(2
5tan
521
13 kk
k
kFkk
k
.273167869100517867... 2
7
1
2 2
1
221
arcsin
kk k
kk
1 .2732395447351626862... 4 2
3 2
1
1 2
5
4
2 3
22
2
2
( )
( / ) /
( )!!
( )!!
k
kk
J274
= 1
2 202k
kk
tan
K ex. 105
= ( )
( )
2 1
2 2 2
2
1
k
k kk
.2733618190819584304... 1
2 121
kk
.27351246536656649216...
1
27
2
3 12
2
31
2
21
( ) log x
xdx
x
e edxx x
9 .2736184954957037525... 86
1 .27380620491960053093... log log( )3
21 3
12
20
1
xdx
x GR 4.295.38
.2738423195924228646... 1
28 2 14 2 13
2 32
1
log log( )
H
kk
kk
.27389166654145381... 1
254 3 3 27 3 2 32 log ( )
=1
31
3
34 31
1
1k k
k
k
kk
k
( )( )
.2738982192085186132... dx
x x4 41
... ( )k
k k k kk k
1
2 1
1 1
22
2
arctanh1
.27413352840916617145...
2 3
31
3
2 21
sin
( )
kk
.2741556778080377394...
1
2
23
cos
36 k k
k
= dxx
x
x
dx
x
1
0
6
13
)1log(11log
2 .2741699796952078083... 32 5 2 71
8
2 3
0
( )
kk
k
k
1 .2743205342359344929... 2 2
11
1 221
coth( /
)
kk
J274
28 .2743338230813914616... 9
...
1
112 2
)1(4
1,
2
3,2,2
2
1
k
k
k
kk
F
.2745343040797586252... 83711
304920
1
111
k kk ( )
.27454031009933117475... k
Likk 5
1
5
1
5
1
20
11 2
/ ( , , )
.2746530721670274228... log
( )
3
4
1
2
1
2
1
4 2 11
arctanhk
k k
=dx
e x20 2
.2747164641260063488... ( ) arctan log
14
1
52 2 2
5
41 2
1
k kk
k
H
7452 .274833361552916627... 9
4
.27489603948279800810... 4
9 62
1
4 31
log
( )k kk
1 .27498151558114209935... 1
3 21k k
k
.275000000000000000000 = 40
11
.27509195393450010741... sin
( )( ) ( )
k
k
ie e e e
k
i i i i
1 221
22
2Li Li
3 .27523219111171830436... 3
G
.27538770702495781532... 2 2
2122
2
21 2
1
21
log
loglog Li
=1
2 121
kk k k( )
... ( )
! ( )
1
2
1
1
k
k k k
.27543695159824347151... ( )
1
13 30
k
k k k k
=1
42
2
2
1
81
21
2
1
2 2
1
2 2
csch
log( ) ( ) ( ) ( )
i i i i
.27557534443399966272... 1 1 1 2
2 11
1
1k k
k
kk
k
k
arctan( ) ( )
( )
1
x
... log tanx x d0
1
.27572056477178320776... cschB
21 1
2
2 2 1
22 2
2 2 12
1
e e k
k kk
k
( )
( )! J132, AS 4.5.65
.27574430680044962269... loglog(cos ) log( cos )
( )cos
21
2
2
81 1
1
4
k
k
k
k
... 4/
0
23
2
cot)(2
1)1log(
162
)3(
4
dxxxiLii
G
... Levy constant
... 8
5
2
5 5
4
5
4
5 1 5 20
csc /
x dx
x 4
.27591672059822730077... ( )( )
( )
( )
1 11
12 2
k
k kk
k
k k
... 1
4 2 22
1
2 3
1
21
csch
( )k
k k k
.27614994701951815422... 1
8 12 3
1
9 421
kk
.27625744117189995523... 5
324 3
3 2
3 30
log x
x xdx
1 .27627201552085350313... 13
32
3 3
50
x x
xdx
sin GR 3.787.3
.276326390168236933... Erfk k
k
kk
1
4
2 1
4 22 10
( )
!( )( )1
.2763586311608143417...
( ) ( )
( )
3 1
6
1 13 3 3
1
k kk
... log( )
25
12
2 1
4 21
k
k k kkk
.27657738541362436498... ( )
!
1
1
1
1
k
k k
2 .27660348762875491939... 2
2 3 222 0 1
0eI e e
e
k k
k
k
( ) (; ; )!( )!
F
...
0
2/)1(
02/3 2
)1(
2
1
2)14(
1
4
1,
2
3
8
1
kkk
kk
k
[Ramanujan] Berndt IV p. 405
... ( )
13
1
1
k
k k
k
.27700075399818548102... 43
234
9 13
23413
csc
= ( )
1
1324
k
k k
.2770211831173581170...
2 1 12
12
1 1
1
( ) ( )! ( )
( )!
k k
kk
k
k
...
0
2sinlog565log32arctan8100
1
25
14dxxxex x
.27708998545725868233... 1
4 12k
k k( ( ) )
.2771173658778438152... ( ( ) ) kk
1 3
2
4 .27714550058094978113... 1
22
1
2, ,
.27723885095508739363... H
k
H
kk
kk
kk
kk4
1
321
1
1
( ) (2)
.27732500588274005705... log2 2 .27734304784012952697... ( )1
3
... x x
edxx
log( )1
10
1 .277409057559636731195...
12 23
1
4
3log3
12
3
k kk
= dxx
x
1
03
3)1log(
.27750463411224827642...
1
2 )!1(
)1(1)1(
k
kk
k
BeLi
1 .27750463411224827642...
0
2 )!1(
)1()1(
k
kk
k
BeLi
.2776801836348978904...
8 2 2 12 20
2
4 20
dx
x
x dx
x( ) ( )
192 .27783871506088740604... 6
5
1 .27792255262726960230... 2 2 2 2sin log ( ) i i i
.2779871641507590436... log7
7
... ( )
( )
1
2 1
1
21
kk
k
H
k k
.27818226241059482875... 12
2
1
2 1
21
1
arcsinh ( )k
k k
k
k
... ( )
!( )!
2
1
1 2
1 11
k
k k kI
kk k
... 1
164 3
1
2 1
2
20
1
, ,log
/
x
x 4
=
3
23 34
2 22 2
log i Li
iLi
i
.27865247955551829632... 11
2 2
1
2 21 1
log kk
x
dx
17 .27875959474386281154... 11
2
.2788055852806619765...
1
1,2
1)11(
xe
dxerf
x
... e
e x
dx
x2
5
22
1
14
sinh
.27892943914276219471... 5
4
5
4 51
log
Hkk
k
... k kk
log ( )
2
.27918783603518461481... ( ) ( )
logk k
k
k
kk
kk k
1 11 1
1
22
1
.27930095364867322411... 33
2
1
6 1 6
1
3 2 320 0
k kk
k
kk/
( )
( )
.27937855536096096724... S k k
k kk
22
1
2
2
( , )
( )
.27937884849256930308... 1
2
1
21
1
2
3
2
1
2 2
1
1
, ,( )
k
k k
=1
22 2 2
1
2
( )
.2794002624059601442... 1
4 1
1
4 11
1
1k
k
k
kk
( )
.27956707220948995874... 1
22
1
2 2
20
1 2
0
Jk
k
k
k
k
kk
( )( )
( )!
2 .2795853023360672674... Ik
k
kk k0 2
0
2
20
21
( )( !) ( !)
LY 6.112
= 0 12
0
2
111
2F e(; ; ) cos
d Marsden p. 203
= 1
2
2
2
2 1 2
0
2
0
2
0( )! ( )!
( )
!k
k
k
k
k
k
ke
k
k
kk k
k
k
.2797954925972408684... 1
102 1 5 1 5
1
531
( ) ( )i ik kk
1 .2798830013730224939... ei
e ek
ke e
k
i icos sin(sin )sin
!1
1
12
GR 1.449.1
1 .27997614913081785... 2 2 1
22
1
1
erfi( ) ( )
!e
kk
k
k k
k
8
=
7
25
1
1
6
2 3
p
pp prime
.28010112968305596106...
41
2
0
2
ee
x
edx
x( )
sin
7 .2801098892805182711... 53
.28015988490015307012... ( )k
k
kk 21
.2802481474936587141...
3
1
61
3
21
3
2
1
2 331
i ik kk
.2802730522345168582... 1
63 9 3 2
2
3
1
3 11
21
log ( )( )
( )
k kk
.28037230554677604783... 5
32 2
3
2
1
2 22
log hgk kk
=( ) ( ( ) )
1 1
2 12
k
kk
k
1 .28037230554677604783... 8
32 2
3
2
13
22
log( )k kk
.28043325348408594672... 1
36
1
3
1
6 41
21
( )
( )
kk
1 .28049886910873569104... 1
2111
1
2
1
2
1
1620 k
kk
HypPFQ { , , },{ , },
... ii eei
2log2log2
1sin)1(
=
2
1)(sin
k
kk
k
6 .28063130354983230561... k kk
( ( ) )2
2
1
.2806438353212647928... 2 2 21
22 log log
l Berndt 8.17.8
.28086951303729453745... 352
735
2 2
7
1
2 71
log
( )k kk
...
1 2!
)1()21(
2
1log
kk
k
k
ke
=
1 2!
)12()1(
kk
kkk
kk
B [Ramanujan] Berndt Ch. 5
.2809462377984494453...
64
27
1
2
3
428 3
64
27
12 33
43
1
( ) ( )( )kk
.2810251833787232758... 2
3 2124
25
192
1
4
k kk
... 2 31
22
1 22
2 1
1
1
log3
2
( )
( )
k k
k k
kk k
... 33 1
3 1log
.28128147005811129632... 1
22
1
2
1 1
1
log( cos )( ) cos
k
k
k
k GR 1.441.4
.28171817154095476464... 320
ek
kk ( )!
=k
k k k k kk k!( ) !( )( )2 6
1
2 30 0
1 .28171817154095476464... 42
2
0
ek
k kk !( )
.28173321065145428066... e
e
e
e
k
k
k
k
1
2
2
2
12
2
1 2
1
cos(sin ) ( ) sin
!
cos
cos
.28184380823877678730... 4
3
8
7
82 2
8
3
8cot cot log log sin log sin
=( )
1
4
1
21
k
k k k
.2819136638478887003... log( )
224
3 1
2 2 1
2
2 21
k
k kk
=( ) ( )
1
2
1
3
k
kk
k k
.28209479177387814347... 1
4
.28230880383984003308... ( )
1
3
1
1
kk
kk
H
k
(3)
.282352941176470588 =24
85
4
0
sin x
ex
1 .28251500934298169528... 3 6 21
1 230
log( )
( )(
k
)k k k
...
2 3
22
2
3
8 2
32 2 3 4 2
1
43 3
log loglog log log ( )
Li
4
1
43Li
=log ( )
( )
2
120
1 1
x
x xdx
1 .2825498301618640955... 6
2 ( )
.2825795519962425136... 1
21
41 20
Ik
k kk
( )( !)
.28267861238222538196... 1
631 kk
.2827674723312838187... 1
332 kk
.2829799868805425028... 0 11
0
2 22 2
2
1 2
1F (; ; )
( ) ( )
!( )!
J
k k
k k
k
.2831421008321609417...
( )
( )
k
k kk
1
2
.28318530717958648... 2 6 2
0
1
arcsin arccosx x dx
6 .28318530717958648... 21
6
5
6
=1
14
340 ( )( )k kk
= log 1 6
0
x dx
=d e
e
x
x
2 10
2 6
cos
/ dx
6 .2832291305689209135... 2 2 coth
2 .28333333333333333333 = 137
60 5 H
.2833796840775488247... ( )
( )!sinh
2
2 1
11
1 1
k
kk
kk k
.2834686894262107496... 11
31 3
1
1
ek
k
kk
/ ( )
!
5 .28350800118212351862... 2 1 23 4 4
0
/ log
x dx
.28360467567550685407... 3 2 3 33
2 3 1
1
2 11 0
log log( )
( )
( )(
3)
k
k k kk
k
kk k
.2837571104739336568... log( )
log
2 2
11
2
1
22 1
k
k kkk k k
.28375717363054911029... 6
251
6
5 61
logk Hk
kk
.28382295573711532536...
2
24
1
6
49
36
14 2 4 1 2 4
kk
( )( ) ( , ) ( , , )
.2838338208091531730... 1
41
1 1 2
2
1
120 0
1
0
1
e k
x
edx
xdx
k k
kx
( )
( )!
sinh
coth
1 .28402541668774148073... e4
.28403142497970661726... ( ) ( )
( )!
2 2
21
k k
k
2
k
=1
1 11 1
2 22 k kk
coshk
.28422698551241120133... ci ci( )sin ( )sin ( )cos ( )cos( )1 1 2 1 1 1 2 1 si si
=sin x dx
x10
1
6 .2842701641260997176... k Hk
kk
2
1 2
... 2
22 2
2122
3
6
2 3
4
1
22
log
log log loglogLi
1
2
1
2
21
4833Li ( )
=log ( )
( )
2
20
1 1
2
x
x xdx
2 .28438013687073247692... ( ) ( )3 4
... ( )
( )!sinh
k
k k kk k2 1
1 1
2 1
k
1
2 .28479465715621326434... 8
11
3 .28492699492757736265... F
kk
k !!
1
... 2 2
24
log
... 1
21 k kk ! ( )
...
0
24/9
sin2
22 2
dxxe x
.28539816339744830962... 4
1
2
1
4 1
1
21
( )k
k k J366, J606
= sin cos sin cosk k
k
k k
kk k
1
2
1
= x x dxarctan0
1
.2855993321445266580... 11
1 .2856908396267850266... 15
32
1
32 2
4
0
e
e
k
kk
( )!
.285714285714285714 =2
7
.285816417082407503426...
13
2
)3(
1
162
11
543
)3(
k kk
1 .2858945918269595525... ( )!!
1 12
1
k
k
k
k
.28602878170071023341... 2
21
2
16
3 2
4 2
8 1
4 4 1
log
( )
G k
k kk
=k k
kk
( )
42
.2860913089823752704... ( )3 G
.28623351671205660912... 2 1
3 2124
1
8
1
2
( )k
k k k
.2866116523516815594... 6 2
16
CFG D1
.2868382595494097246... 1
4HypPFQ { , , , },{ , , },
( !)
( )!( )1111
1
22 2
1
4 2 2 2
2
20
k
k kk
...
4 2 2 8 2
2
9
2
2 1
22
2
22
coth( )
cschk
kk
= ( )( )
12 1
21
1
kk
k
kk
1 .28701743034606835519... e erfk k
k
1 32
1
12
1
4 2
1
4/
!!
1 .287044548647559922... 11
22
( !)kk
... e k
k
( )
1
2
1
1 .2872818803541853045... 2 17
18
3
3
3 2
1
e k
k
k
k
( )!
.28735299049400502... Patterson Ex. 4.4.2c4
1
12010 2 20 3 15 2 30 4 24 55 3 2 2( ( ) ( ) ( ) ( ) ( ))
2 .2873552871788423912... e
k
k
k
k
sinsin
!
/
12
42
1
...
2
1
2
1
4
2log2 22
iLi
iLiiG
... 2 11e ee ei icos cos(sin )
e
3 .2876820724517809274... 2 2 31
41 1
1
41
log log ( , , )
kk k
J117
=dx
x x
dx
x x
dx
e
dx
ex x21
2
21 00
2
3 1
log
3 1
=( )
( )
1
3
1
42
1
7 2 12
1
7
1
11 2 1
0
k
kk
kkk k
Li arctanh
10 .287788753390262846... k
k
e
k !
0
1 .28802252469807745737... log1
4
2 .288037795340032418... ( )
.2881757683093445651... 6
5
1
5
1
51
1
521
1
1
hgk k
k
k
kk
k
( )( )
3 .28836238184562492552...
1
3
!
)(
k k
kt
.28852888971289910742... i i i
kk6
2
3 3
2
3 3
1
3 2 20
( ) 1
.2886078324507664303... 2
2
0
( )e edx
xx x GR 3.463
( )ex
dx
xx 2 1
120
GR 3.467
.2886294361119890619... log
( )(
2
5
3
20
1
2 1 2 60
)
k kk
.2887880950866024213... 11
21
kk
=
12/)3(2/)3( 22
2
1
2
1)1(1
kkkkk
k Hall Thm. 4.1.3
.28893183744773042948...
42
0
2
ex x x d sin(tan ) sin tan
/
x GR 3.716.8
.2889466641286552456... 1
4 2 11k
k k( )
.2890254822222362424...
201
sin x
edxx GR 3.463
.289025491920818... 1
one ninth constant
1 .28903527533251028409... 4
3
1
3 2 1
3
2 2 120
2
20
( )
( )
( !)
( )!( )
k
kk
k
kk
k
k k
=
3 3
310
275
1
3 1
2
20
log( )kk
Berndt 32.7
=
3 33
10
27
5
9
1
3
21log ( )
=
4
3,2,2,
2
1,1,1,1,1HypPFQ=
2
1,2,
3
1
3
1
5 .28903989659218829555...
1
5
... ( )
log
12
2
k
k k k
.289144648570671583112... ( )
1 1
1
k
kk F
.28922492052927723132... 2 3 3
48
1
3 3 10
( )
( )(
k
k k k )
2 .2894597716988003483... 2 log 2 .2898200986307827102... li( )
...
1001
3455
1
1
2 4
2 2
p p
pp prime
.2898681336964528729... 2
2 2113
31
1 1
22
k k
k
k kkk ( ) ( )( ) K Ex. 111
= k kk
( ( ) )
3 11
1 .2898681336964528729...
2
232 2 2 2 1 1 1
( ) ( ( ) ( ) ( ) )k k k kk
= 11
12 21
k kk ( )
2 .2898681336964528729...
1
2
2)2()1(13 k
kkk
= dxe
ex
x
0 1
1 GR 3.411.25
=
1
10
1 x
xx dxlog GR 4.231.4
3 .2898681336964528729... 2
233
2 21
2
( )( ) (k k
kk
)
=x dx
x
x dx
x
x dx
x
2
2
2
20
2
211 1sinh
log
( )
log
( )
GR 4.231.4
=log ( ) log( )2
20
1
0
11 1
x
xdx
x
xdx
=x dx
e ex x
2
0 2
=
0 13/1xe
dx
.28989794855663561964... 6 1
5
CFG C1
.2899379892228521445... 11
1 11
1
1
1
e
eH
e
kk
kk
( log( ))( )
( )
57 .289961630759424687... cot cot1180
.290000000000000000000 =29
100
1
7 621
k kk
... ( )
1
2
1
1
k Ok
kk
H
... 2 21 1
0
/ (e k
k
k
) / !
.29051419476144759919... sin x dx
ex
20
=1
41 1
1
211
1
22 22 1 2 1F i i F i i, , , , , , ( ) cos(log( ))
csch
2 .29069825230323823095... e erfkk
( )( )
11
120
!
.2907302564411782374...
4 3
3
2
1
6
1
4 321
coth
kk
.29081912799355107029... 4 81
2
1
4 2 30
arctan( )
( )
k
kk k
... 1
2 2
1 1
1 10
1
e
e x
e xdx
log( ( ) )
( )
.291120263232506253... 2 2 2
20
1
24
2
4
1
1
log log( )
( )
x dx
x x GR 4.295.17
=
log( / )1 22
0
1 x
xdx
.29112157403119441224... 1
5 1
3
531 1k
k
kk
k
( )
.2912006131960343334... 2 2 2 2 2 12
2 1 11
1
1
cos sin ( )( )!(
kk
k
k
k k )
.2912758840711328645... 1 1 2
12
1
2
1
42 2
1
1 1/( )coth
kk k
dx
x
1 .291281950124925073115... 3
24
1 .29128599706266354041... 1
1 0
1
k
dx
xkk
x
GR 3.486
5 .2915026221291811810... 28 2 7
.29156090403081878014... G
.29162205063154922281... 1
4 16
2
4
7 3
8 1
2 2 3 2
20
1
log ( ) arccosx x
xdx
.29166666666666666666 = 7
24
1
2 1
1
214
0
1
( )
( ) (
k
kk k
dx
x )
.29171761150604061565...
( )2
31
kk
k
.29183340144492820645... 3 4 13
314
2
k
k k
kk
( ( ) )
.2918559274883350395... ( )k
kLi
k kk k3
23
1
1 1
.2919265817264288065... cos ( )( )
2 12 1
1
1 1 12
2
!
kk
k k GR 1.412.2
= ( )
( )!( )
1 4
2 1
1 2
1
k k
k
k
k k
... 3 2
2124 1
arctan x dx
x
.291935813178557193... ( )( )
!/
11
11
2
1
2
k
k
k
k
k
ke
k
.292017202911390492473... 3
4 2 0
1
G
x x xarccot log dx
.29215558535053869628... ( )
!( ( ) )
1
12
k
k k k
.29215635618824943767... 1
2 6 2
12
3 2
2 21
coth
( )
/
k kk
.29226465556771149906... ( ) ( )2 1 2 1 11
k kk
.292453634344560827916...
( )log
k
kk
1
1
1
23 2
2
=
kk kk
log 11
11
22
.2924930746750417626...
2 5 5
1
2
1
5 121
coth
kk
.29251457658160771495... 1
844
2 2
1
e ek k
kk
cos cos(sin )sin cos
!
.29265193313390600105... H
kk
kk
( )3
1 4
.2928932188134524756... 11
2
= ( ) ( )( )!
( !)
11
4
21
2
41
1
12
1
kk
k
kk
k
k
k
k
k
.2928968253968253 = 7381
25200 10
1
10
1
2510
21
26
H
k k kk k
3 .293143919512913722... 1
2
1
2
3
21
2
3 2
1
( , , )/
kk
k
3020 .2932277767920675142... 7
.29336724568719276586... log( )1
1 30
1
x
xdx
.29380029841095036422... sinh
cosh1
4
14 5
1
x
dx
x
1 .2939358818836499924... ( ) log ( )k kk
2
.2941176470588235 =5
17
.2941592653589935936... 10
51
50
1
2521
coth
kk
J124
.29423354275931886558... i
i ix x
e ex x22 2
11 1
0
( ) ( )( ) ( )sin
... ( ) ( )
log
1 1 11
1 12
2 2
k
k k
k
k k
k
k k
k
.29430074446347607915... ( )
1
2 11
k
kk
.2943594734442134266... Ik
k k kk
20
21 2
( )!( )!
.2943720976723057236... k k
kk k
k k
6 3
3 32
2
113 1
( )( ( ) )
.29452431127404311611... 3
32
33
22
1
k k
kk
( )
( ) J1061
.2945800187144293609... 412
2 21
2
2 1
3 21
log
( )k
k k k
.29474387140453689637... 1
4
2 1
12
( )
( )
k
kk
.29478885989463223571...
22
2
)(
111
)(
)1)((
kk kk
k
= 112
( )
( )
k
k kk
2 .2948565916733137942... ( )kk
2
... e2
4 .29507862687843037097... k
kkk
3
1
.2954089751509193379...
6
... 2 2
31
14
1
log log( )
x
xdx
= dxx
x
1
0
2 11log
3 .295497493360578095... e
.2955013479145809011... 11
12
22
12
2
( )( )k k
H
k
.2955231941257974048...
( )( )
11
k kk
57 .2955779130823208768... 180
, the number of degrees in one radian
.29559205186903602932... ( )( )k k
kk
1
41
... ( )
1
4 4
1
21
k
k k k 5
.29583686600432907419... 3 3 3 31 3
0
log
ee
x
dx
xx
x
GR 3.437
3 .29583686600432907419... 3 33
1
1 2 3 1 3 3 1
0
1
log/ /
x x x x
xdx
n n n n
GR 3.272.2
...
log log
( )( )2 2 2
21
2
2 12
2
123
1
2
k
kkk
.29593742160765668... 10 14 22 1
2
1
2
log( )(
k
k kk )k
5 .2959766377607603139... cosh
sinh( )!
1
2 2
2
4 42
4
0
e e
k
k
k
.29629600000000000000 = 37037
125000, mil/mile
.2965501589414455374... 9 3
121
13
0
1
3
x
xdxlog
.29667513474359103467... 11
21
k kk
.29697406928362356143... 7 312
4
12
41
( )
( )
k
kk
.29699707514508096216... 1
2
3
21
2
121
0
e
k
kk
k
k
( )( )!
.297015540604756039... 1
2
3
12
3
2
1
4 421
2
tan
k kk
.2970968449824711858... 2 2 23
21
1 2
2 12 1
1
1
Fk k
k
k
k
, , ,( ) ( )!!
( )!!
.2974425414002562937... 2 21
1 20
ek k
k
( )!
.2974425414002562937... 22 1
2 20 0
ek
k
pf kk
kk
k
!
( )
.29756636056624138494...
0
1
246
1
612
)1(
k
k
k
.2975868359824348472... 2
22
206
31 5
2
1
2 2 1
log( ) ( !)
( )!( )
k
k
k
k k Berndt 32.3
.29765983718974973707...
0
3
3
3
4
3
1xe
dxx
...
1
1
2)1(
1k
k
k
k
.2978447548942876738... Hk
kk
3
1 6
.29790266899808726126... 1
0
2arctan)223log(224
1dxx
.2979053513880541819... H
kk
kk
(2)
41
.29817368116159703717...
e
Eixe dx
x
x
21
1
2
20
( )
.29827840441917252967... 1 2
22 k
k
kk
log
.2984308781230878565... 1
21
logk
kk
... 2
3 29
2
2
20
1
log( )log( )
x
xdx
... ( ) ( )
1
1
1
2 12
k
kk
k
j kjkk k
3 .2985089027387068694... 32
945
49
, volume of the unit sphere in R
.2986265782046758335... log6
6
.29863201236633127840...
0
2
)!3(
2
8
5
8 k
k
k
e
.2986798531646551388...
3 33
3 3 13 230
log( )
x dx
e x GR 3.456.2
=
x x dx
x
log
( )1 3 230
1
GR 4.244.3
.29868312621868806528... ( )( )
12 1
11
1
k
k
k
k
.2987954712201816344... 15
1624 3
3 3
8
1
3 41
log
( )k kk
1 .29895306805743878110...
0 22
1
22
1arcsin
77
8
7
8
k k
k
k
1 .2993615699382227606... ( )k
kk2
2 2
.29942803519306324920... ( ) cot2
1
2 2 2 2
1
2
2
4 21
k
k kk
= ( ) ( )2 2
21
2k kk
k
.29944356942685078... ( )
1
532
k
k k
.2994647090064681986... 6
3 3e e J132
8 .29946505124451516171... 1
211 1
0
e e ek
ke e
k
/ cosh cosh(sinh )cosh
! GR 1.471.2
2 .2998054391128603133... 3 1 12
120
logx
dx
.299875433839200776952... 2 2
4148 3
1
3
2
4
G x
xdx
log arctan
5 .29991625085634987194... ( , )1
3
1
3
1
32
3
2
.300000000000000000000 = 3
10
1
2 31
0
2 1 3
cosh log( ) ( ) logk
k
ke J943
= sin cos3
03
0
x
edx
x
edxx x
.300428331753546243129...
1
0
12
)()(
kk
kk
.300462606288665774427... 12
13
.30051422578989857135...
( )
( ) log log3
42 3
2
32 2
1
22 2
1
23
2 3
Li Li
[Ramanujan] Berndt Ch. 9
= log log
( ) ( )
2
31
2
1 1 1
x
x xdx
x
x x xdx
= log2
1
2
1
x
xdx
[Ramanujan] Berndt Ch. 9
= x dx
e x
2
20 1
= log ( ) log( ) log2
0
1 2
0
11 1
x
xdx
x x
xdx
= logsin tan/
x x2
0
2
dx
2 .300674528725270487137... 2 3 ( )
.300827687346536407216... H
kk
kk 2 2 11 ( )
1 .3010141145324885742... ( ) ( )
( ) ( ) ( )3 4
3
90
41
41
13
1
k
k
k
kk k
HW Thm. 290
... log10 2
... 0
1
( )
!
k
kk
.30116867893975678925... sin cos1 1
=
1 1
11
)!2(
)12()1(
)12()!12(
)1(
k k
kk
k
k
kk
= 3
1
1
0
1sinsin
x
dx
xdxxx
= dxx
xxe
1
logsinlog
... 2 11
11
1
1
4
20
20
log logx
xdx
xdx
=
022 )1)(2/1( xx
dx
2 .3012989023072948735... sinh Re{sin(log )} cos ( ) 2 2
1
i i i
= 11
2 1 20
( )kk
.31034129822300065748... 2
12sin
1
)1(
2
1cos
2
1sin)1cos22log2(
2
1
1
1
k
kk
k
.301376553376076650855... 6
2
92
0
1
x xarcsin dx GR 4.523.1
... 12
21
2
2 11
2
1
sin( )
( )!( )k
k
k
k
k k
... 1
5 1 51
0
1k
kk
k
k
( )
... log log( )2
0
13
4
1 2
1 2
x
xdx
... 22
22arctane g d
1 .3018463986037126778... log ( )( )
loge e k
k kk
k k
2
12
111
12
1
= logsinh
log log
1 1i i
... Ik
k
k0 2
0
44
( )( !)
... 840
3 3
2 1
161
k
k
kk
k
... 1
2
1
2
2 1
2 11
logcotcos( )
k
kk
GR 1.442.2
...
0 )33)(13(
)1(
36 k
k
kk
= dx
x
x dx
x x
dx
x30
4 21
2 28 1
( )3
=
012
7
012
3
11 x
dxx
x
dxx
...
2 2
1)(
kk
k kH
2 .302585092994045684018... log 10
...
03
43
)2(3
)(
12
4
32)3(56
k k
=
334
)()2)(1(
kk
kkk
... 2
2
0
1
122
2
1
loglog( / )x
xdx
... log2
3
... 1
2
1
2
1
1e k
k
kk
( )
!
... 2 5 5
2
5
3
5csc sec
... 11
2
1
2
1
2
cos
sinsinh
coscosh
cos
= 1
1
21
21 21
1
cos
sin
( )cos
!(cos ) /e
k
kk
kk
... 4
4 11
1
2
20
2
2 2(log ) log( )
x
x
x dx
x GR 4.298.19
... log!
11
1
kk
... 8
9 3
2 2
3
1
3 2
1
1
log ( )
( /
k
k k k )
...
2 2 2
5
61
2 1
21
1
coth ( )( )
kk
k
k
= 1
2 122 kk
... 21
3
1
2 23 3
2
0
1
Li Lix
x xdx
log
( )( )3
... 2
2 2
0
1
12
14
27 x xarcsin dx
.30396355092701331433... 3
2
... 3 2 21 2
12
1
( )
( )!
k
k kk
!
... 24
2
4
1
16 1221
log
k kk
... 1
2 2
1
22 2log
sinhlog ( ) ( )
i i
= ( )( )
12 1
21
1
k
k
k
k
... log log log ( )2 2 2 22
3
x dx GR 6.441.1
.3053218647257396717... G
3
... 5040 185471
ek
k kk !( )
... 1
23 kk !
.30562962705065479621... 3 2
2
1
31
2
2 1 3 2 11
arcsin( )!!
( )!! ( )
k
k kkk
... 1
3 22 k kk log
... Fk
kk
3
1 6
... 63
2 3 12
2
0
1
( ) log( ) logx x dx
... ( )
( )!
/k
k
e
kk
k
k
2 1
11
1
32
.30613305077076348915... 4 3
27
1
2 12 30
dx
x x( )
1 .30619332493997485204... 3 2
4
3 2
2
3
22
1 4
23 2
3
1
sin cos( )
( )
( )!/
J
k
k
k k
k
13 .30625662804500320717... 128
9
1 3 1 1
42
1
Gk
kk k
kk
( ) ( )( )( )
... 1
4
3
42
0
4
x e dxx
1 .306562964876376527857... 1
24 2 2
8 82
5
8 cos sin sin
= 11
4 20
( )k
k k
... 12
3
1
30
1
21
Jk
k
kk
( )
( !)
8 .3066238629180748526... 69
... 26
405
6
1 11 1 1
4
1 3 41 3 1 3
42 3
( )
( ) ( ) ( )// /Li Li /
= log3
3 21
x
x x xdx
...
122
22
)1(
1
2
1csch
4coth
4 k k
... 1 21
122 2
log( )
( )( )k
k
kk
kk
k
= k
kkk
1
22
J145
= 1
2 11k
k k k( )
J149
= 1
4 22
1
2 21
1
31k k k kk
k
k( )
( )
( )
= dx
x x k k
dx
x xk3 2
12
12
1
1
= dx
e ex x
10
= e ee
x
dx
xx x
x
2
2
0
GR 3.438.3
= ( ) /x e edx
xx x
1 2
0
GR 3.435
= xx
x
dx
x
1
0
1
log log GR 4.283.1
= xe
edx
x
x
20 1
GR 3.452.4
= log x
x xdx
2 21 1
GR 4.241.8
= GR 4.384.5 log(sin ) sin/
x x d0
2
x
x ... GR 4.591.2 2 20
1
log arccos logx x d
... 3 1 2
4
1
4 2 1 2 3
2
0
arcsinh
( )
( )( )
k
kk k k
k
k
= x xarcsinh0
1
dx
.307654580328819552466... 3 3
8 48
1
16
1
2
2 1
31
( ) ( )
( )
k
k k k
.307692307692307692307692= 4
13
.307799372444653646135... 111
1
1
ek e
ek
kk
/ ( )
!
.30781323519300713107...
log(cos )( )
sin1
21 1
2
1
k
k
k
k J523
... arccot
( )
( )
1
2 12 11
k
kk k
... cosh( )
42
16
2
4 4
0
e e
k
k
k ! AS 4.5.63
.308337097266942899991... ( )( )
13 1
1
1
kk
k
k
.308425137534042456839... 2
20
2232
1
4 2
1
2
( )
( ) (
k
k k
kk
k
)
= x x
xdx
x x
x
x
x
log log arctan4
0
1
41
20
1
1 1 1
...
0 03
1
20
2
!
)1(
)!(
)!2()1()2(
k k
k
k
k
kk
k
e
I
...
1
0
1
0
1
0
2
12log6
43 dzdydx
xyz
zyx
...
022
)1(
k
k
k
... 4
... 5
12
...
Li2
1
3
= 1
23 2 3 4 4 2 2
1
42 2
2log log log log
Li
= 1
63 3 3 4 6
3
42 2 2
2
log log Li
= ( )
1
3 4
1
21 1
k
kk
kk
kk
H
k
... 3
2 6
9 3
29
1
3
2
3 21
log
k kk
= ( )( )
12
31
1
kk
k
k
... 24
3 CFG A10
...
1
...
0
2
2
2
)2log(
22
2log3
x
x
... 6e
.309859339101112430184... 1
2 2 6 6
1
6 1
2
621 1
cot
( )
k
k
kk
k
... )()( ii
... log
( ) ( )2
2
1
8
1
2
1
2
i ii i
= sin2
0 1
x
edxx
... ( )( )
( )( ) log
12 1
11 1
111
1
22
2
k
k k
k
k kk
k
... k
Hkkk 21
...
2 2
23
2
3 8
3
8 42 3 coth coth ( )csch csch
= ( ) ( ) ( )
1 22
2
k
k
k k k
... 9
1 116
1
17
1e
k
k
k
kk
k
k
k
( )!
( )!
...
159
1820 2 1414
1
8 221
cot
k kk
... 1
2 2
3
41
32 2
0
2
sin
/
x dx
=
5
43
41 4
0
1
dx
x
... 1
0 k kk ! !!
... 2
11
2 20
e
xe x
dx
log log
... 2
41arcsinh
.311696880108669610301... 22 sinh2
csch16
1
=
122
2
1
1
)14(
4
4
)2()1(
kkk
k
k
kk
.31174142278816155776... sinsin
coshcos
sinhcos
sinsin
(cos ) /
1
2
1
2
1
2
1 1
21 2
e
= ( ) sin
!
1
2
1
1
k
kk
k
k
.311770492301365488... ( )( )
!
11
1
1
k
k
kH
k
... 2
3 3
1 3
2
3
3 3
1 3
22 2
i
iLi
i i
iLi
i
= 5
36
1
6
1
3 1
21
20
1
( ) logx x
x xdx GR 4.233.4
= x
e e edxx x x( )
10
GR 3.418.2
... 7
7
2
7
8
1
5 821
tanh
k kk
... 7
.311993314369948002... 12
3 2
22
2
2
2
log
loglog
=( ) ( )
1 1
11
k
k
k k
k
= 1920
14631
1
71
1
111
1
19
= 2 11
31
1
71
1
111
1
192 2 2
2
[Ramanujan] Berndt Ch. 22
... 3 3
4 62 4 2 8
1
2 1
2 1
31
( )log
( )
( )
k
k k k
... 2 1111 2 2 2 2 22 3
1
HypPFQk
kk
k
{ , , , },{ , , , ,},!
= 5
16 5
1
2
1
113 1
23 2
2
k
k k k k kkk k k ( )2
= k kk
( )2 1 11
2 .3126789504275163185... ( )2 1
21
21
2
k
kLi
kk k
... ( )
! ( )
1
1
1
1
k
k k k
... 2 3
13 2
e
x
xdx
cos
( )2
.31284118498645851249... log ( )( )1
22
22
12 1
21
1
i ik
kk
kk
.312941524372491884969... 12
1
6
2
6 41
log arctan x
xdx
... (coth )!
1 12
1
2 22 2
1 0
e e
B
kkk
kk
k
... coth cot( )!
11
1
4
2
2
22
0
e
ei i
B
k
kk
k
AS 4.5.67
... 2
11
22
21
0
e
e
B
kk k
k
k
( )!
... arccsch
... log arctan( )(
34
10
3 2 9 12 21
k
k kk )
[Ramanujan] Berndt Ch. 2
... 1
531 kk
... 1
11 1
2 21 1e e k e ke
k
ekk
k
( )
.313261687518222834049... log( )
11 1 1
1
e e
k
kk k
=
1 1xe
dx
1 .313261687518222834049... log( )1 e
...
( )
( )
3 1
2 1
.31330685966024952260... )2(2
1)2(
3
24,3,
2
1F
2
1120211 I
eI
e
=
k
k
kk
k 2
)!2(
1)1(
0
.3133285343288750628...
1
211
)!1(
)!()1(
24
1
32 k
k
k
k
=
2/
04
2tan2
cotcos
cos22
dxxx
xex x GR 3.964.2
.3134363430819095293... 23
42
1
4
1
1 31 1
ek k k kk k!( ) ( )! !
.313513747770728380036... 2 23
23
3
2
1
6 21
log log
k kk
= dxx
x
1
02
6 )1log(
...
1
21
31
3
1
3 31
i i
k kk
=
1
1
3
)12()1(
332
1
kk
k kii
.313666127079253925969... log( )
( )
2
3
... ( )
! (
1
2 2 1
1
1
k
kk k k )
Titchmarsh 14.32.3
... 1
216 3
1
4 1
2
40
1
, ,log
/
x
xdx
16
... 3
3
2
2
3
1
3
hg hg
2 .31381006997532558873... 4
12 372
2 21
2 1
1 1
log( )H k
k k kk
k
...
( )
( )
( )
( )
( ) log2
2
2 2
22 3
2
21
k k
kk
1 .31413511119186663685... 3
4 16
2 4 3
30
x
xdx
sinh
... 10
... 7129
22680 9
1
99
21
H
k kk
... 1036
225 3
22
5 2
H ( )/
... H
k kk
kk
( )
( )
3
1 2 1
... e e Eik
kk
( )( )
( )1 1
1
11
!
... ( ) ( )
1
2
1 3
1
kk
kk
H
... 1
3
1
3
1
3 2 1
1
1
sin( )
( )
k
kk k !
...
0
)2/1(
11
2
3
2 xe
dxxi
... 1
1 51 ( )
kk
... e2
... 3
22
2016 4
24
loglog x
xdx
... ( )( )
( )
13
2
2
2 11
3
3 211
kk
kk
kk k
k
...
1
9 332log
xx
dx
... 3 3
6 3
6 3 3
6
3 3
6 3
6 3 3
65 6
7 6 2 3
5 6
7 6 2 3
i i i i/
/ /
/
/ /
3
3 31
1
35 6 1 3/ /
= 1
3 11
3
321
11 k
kkk
kk
( )( )
... 9
8
15
161
2 51
eerfi
k
k kk
!( )
...
( )log ( )
k
kk
k
2
=6
19
.3158473598363041129... tan x dx
ex0
1
... log( log( ))1 10
1
x dx
... 1
2
1
2
2
0
1
ex e dxx
=
13
152
1cosh
2
11cosh
x
dx
xx
dx
x
... 31 4/
... 10
10
.3162417889176087212... log (loglog
) log (log
)2 32
23
3
2
= ( )( )
121
kk
k
k
k
...
loglog
log( )3
2
1
30
1
1
x dx
x
k
kkk
GR 4.221.3
... 3 3 2 2 ( ) ( )
.31642150902189314370...
1
2411
2 14
)24(
14
)2(
2
1
kk
kk
k
kkk
.3165164694380020839... 1
41
40
2
21
Ik
k kk
( )( !)
... Ei e Ei e dxex
( ) ( ) 10
1
... 11
.316694367640749877787... 2 2 2 2 22 1
81
( log log( )
k
k kkk
=( )
( )!
2 1
2 21
k
k kkk
!!
... 1
3
1
3 3 41
e
dx
x
e
... cosh sinh
( )
1 3 1
16 8
1
16 2 1
4
1
e
e
k
kk !
... arccosh arcsinh2 21
22 3 log
= log3 1
3 1
.3171338092998413134... 1 2 1 2 21
12
1
( ) log log
( ) ( )
( )
k
k
k
k k
... 3
... 12 2
sinh /
... 13
48 24
11
122
1
2 2
50
1
log
log
( )
x
xdx
... e i i3 2 33
2
3
2 / cosh sinh
... 67
120 1
4 3
3 21
3
0
log
( )
x
x xdx
x
e edxx x
.31783724519578224472... 3 2
1 .3179021514544038949... eEik k
k H
kk
k
k
( )! (
11
11 1
)!
=1
11 1 0
1
( )! !loglog log
k kx dx e x dx
k
e
x
... 1
211
1
211
1
2
1
2 12 1 2 1 20
F i i F i ikk
k
, , , , , ,( )
...
2
3
3 3
2 2 3
log GR 8.366.7
... 1
1 11
k k
k kk
( )
( )( )
x
J1061
= x x dsin0
1
... 1
83 22 2 3 2 coth coth csch csch
1
81 11 1 2 2i i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( )1 i
1
= ( ) ( ) ( )
1 2 21 2
1
k
k
k k k
... cosh
( )
5
14
2 1 21
kk
5 .318400000000000000000 = 3324
625 4
2 2
1
F kkk
k
... ( )1 7
6
... ( )1 1
6
... 3
422
2
0
2
ex x cos( tan ) cos
/
dx GR 3.716.5
... ( )
12
1
1
k k
k
Hk
k
.319060885420898132443... ( )( )
( )!cos
12 1
21
11
1 2
k
k k
k
k k
... 4 2 114
3
1
2 2 50
arcsinh
kk k( )
... 41 5/
.319737807484861943514...
log log( )
( )2 2
1
2 12
1
k
kkk
... arccot cot csc ( )sin
1 2 1 12
1
1
kk
k
k
k
... logk
k kk3 2
2
... 2 2
3
1 3
1 3
2
0
1log log( )
x
xdx
... ( )( )
11 1
22
21
k
k k
k
kLi
k k
... 1
11 1
21 ( )!( )
log
!n
k
kk
n
kn
... k
Fkk
1
...
Lik
k
kk
3
1
31
1
3
1
3
( )
... ( )( )
( )!sin
12 1
2 1 2
1 1
21
2 11 2
kk
k k
k
k k
k
... 2
1
4 2
22
1
ie
k
ki
k
logsin //
=i e
e
k
k
i
ik2
1
1
2
21
logsin/
/
...
2
21
4
2
10
cothsin
x
edxx
...
1
2)22()2(k
kk
= 26
81 1 40
2
dx
x( )
.321006354171935371018... e k
k kk
1 4
14 4
/
!
... cot1
2
... 7
48 42
3 2
4
1
1
2 2
21
loglog log( )
( )
x
x xdx
2
... 4
5
2
5 5
2
5
3
5 1 5 20
csc /
dx
x
... 4
3
2
3 32 3
4
31
16
9 1222
2
log ( )
k
k k
kk k
... 2
1
3
1
3 2 1
1
2 10
erfk k
k
kk
( )
! ( )
... 16
63 37
37
2
1
7 321
tan
k kk
.32166162685421011197...
( )
log ( )k
kk
k
1
.3217505543966421934...
1
12
1
)12(3
)1(
42arctan
3
1arctan
kk
k
k
=
1
21
)1(
2arctan)1(
k
k
k [Ramanujan] Berndt Ch. 2, Eq. 7.5
= arctan( )
1
2 1 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
=dx
x21
2
1
... HypPFQk
kk
k
{ , , }, , ,
( )
1111
22
1
4
12
10
... 2 3 4 ( ) ( )
2 .32185317771930238873... 1
2111ijk
kji
... log5
5 4
2
1
F
kkk
k
... log2 5 ... log2 10
... 3
3
2
31
21
1
5 721
tanh
k kk
... F
k
k
k
k
2
1 2
... 2
2212
1
2
1
2
kk
= 2
22
22
24
1
2 Li e Li ei i( ) ( )
= H
k k kk
k ( )(
1 21 )
= ( )( ) ( )
11
21
2
k
k
k k
= ( )cos
1 12
2k k
k
...
12
0 )()(
k k
kk
1 .322875655532295295251... 7
2
... 2 202
0
2
I e x( ) cos dx
... H
k kk
kk
( )
( )
2
1 2 1
... ( ) ( ) ( )( )
2 2 3 41
2
41
k
kk
... ( ) (
!
k k
kk
2
2
)
... 15
8
7
2
1 .32338804773499197945... 9
251
1
3 21
sinh
( )
kk
... ( )
1
3 10
k
kk
... 1
23 2 2 1
11
2
log log ( ) ( )
k
k
k
kk
... sinh( )!
1 1
2 1 2 10
k k
k
AS 4.5.62
... log log( )2
0
15
8
1 4
1 4
x
xdx
3 .3239444327545729800... 121
H kk
k
.323946106931980719981... arccsc
... 1
2 11 2 1
0cosh( ) (
k k
k
e ) J943
.324137740053329817241... 2
2 216 2
1
4
1
2
Li e Li ek
ki i
k
( ) ( )cos
2 GR 1.443.4
... ( )
( )!sinh
2 1
2 1 2
1 1
22 11 2
k
k kkk k
k
...
15
3015
491
770
1
8 121
cot
k kk
... 23 2
0
1
8 21
loglog( ) log
/
x x
xdx
6 .324555320336758664... 40 2 10
1 .324609089252005846663... cosh( )
4
11
4 2 1 20
kk
J1079
... ( )
( )
1
1 3
2
1
kk
k
2 .32478477284047906124...
1 1 11 11
0
1
3
2
1
12
1
12
)(
2
)(
i j kijk
k jjk
kk
kk
kkt
... 3 6
3
1
2
1 12
1
( ) ( )
!
k
kk
k
k
!
... arctan e d
e ex x
2
2 20
1
2 8
x
8 .32552896478319506789...
1
2
12
)(
kk
k
.325735007935279947724... 1
3
... ( )
( )!cosh
2 1
2
11
1 2
k
k kk k
... Hk
k
k
kk 22
1
... ( )
!
k
kk
1
... ( )( )
( )!sin
1 22
2
11 2 1
1
2
1
k k
k k
k
k k
.3264209744985675... H ( )
/2
1 6
... 224
3128
1
16
2 2
0
kk
k
k
... 12
1
2
1
2
1
21
2
21
HypPFQx
xdx, , ,
cos
.32655812671825666123... 1)1(3
)1(62log2
34
1
kk
k
k
1 .32693107195602131840... ( )4 2
2 21
2
41
k k
kk 1k k
... e ( )3
... )(zofzeroofpartimaginary 9 .3273790530888150456... 87
... 17
42 2 6 3
1
2
3 21
( ) ( )log
( )
x
x xdx
.3275602576698990809... 1
32 3 1
21
( log log ) ( )( )
kk
k
k
... 4
3 ( )
... 8 41
1 2
1
1 2
1
2
1
21
Gk k
k k
k
( )
( / )
( )
( / )
... 82
13
41
Gk
kkk
, Adamchik (28)
... 2 56 31
42
13 2
14
30
( )( )
( )
kk
... 1
1 122 k
k
kk
log
... 11
21 1 ( ) ( )i i
= ( ) ( ) ( )
1 22
k
k
k k 1
.32819182748668491245... 1 11
1
22 1 1
1
1 4
2F ( , , )
( )!
k
k
kkk
=( )!!
( )!!( )!
2 1
2 11
k
k kk
... 256
35
4
1 2
/
1 .32848684293666443478... 3 2
2 16
3 3
30
1 log arctan
x
xdx
= 2/
03
3
sin
cos
dxx
xx
... ( )k
kkk 21
13 .32864881447509874105... 23
... 2
22
032 2 2
1
2 2
log( )k
k k
=
0 2/1xx ee
dxx
... 2
30
...
1
0
1
022
2
1
)1log()3(72log2
16
1dydx
yx
xy
... dxxe x )1(2
5
4
3 2
0
2
.32940794999406386902... | ( )| k
kk 41
.3294846162243771650... 1
91 2 2 2 2 3 1
21
( log log ) ( )( )
kk
k
k k
...
3 2
2
1
0
1
loglog
log logx
x xdx GR 4.325.11
... 2
3
39
12
1
1
2 2
0
1
( ) logx x
xdx
... arctanh1 1
2 12 10
kk k( )
... k
k kk 2 2 11 ( )
...
162
23
22
2 22 2 2coth coth
csch csch
= ( )( ) ( )
( )
12
2
2 2 1
2 12
1
2 2
2 31
k
kk
k
kk k k
k
...
2
1
2
1036
225
7
2
( )
... 2 1 11
e Eih k
kk
( )( )
!where h k Hi
i
k
( )
1
... 45 5
2 1
4
0
4
21
( ) log
x dx
e
x
x xdxx
... log( )e 2
... )1sinh1cosh1cosh()1sinh()1cosh(2
1 ee
)1sinh1cosh1sinh(2
1
= e e
e
k
k
e e
k
2 1
12 1
/ sinh
( )!
... 1 4/
...
32
3
1 2 3
3 44
,( )( )( ) (
k k k kk
k
)
... 1
21 1 1 1
0
1
cosh sin cos sinh sin sinh x x dx
.33194832233889384697... 4 4 3 1 32 2
0
dx
x x( )( )
...
2 2
3 3
2
3
2
1
6log
log Berndt 8.6.1
... ( )3 1
2 213
1
k k
kk 1k k
.33233509704478425512... 3
16
1 3
1
1 12
1
( ) ( )!
( )!
k k
k
k
k
.33235580126764226213... ( )4 3
4 14 31
kk
k
... 4 2 .33302465198892947972... log 3 2 ... 2
.3333333333333333333333 = 1
3
= 1
3 3
1
3 1 3 4
1
2 1 2 51 00k k k k k kk kk( ) ( )( ) ( )(
)
= 1
5 6
1
4 421
21k k k kk k
3
= ( )
( )
1
21
22
1
1
1
1
k
kk
kk
k
kk
k
=
11 19
3)2(
13
)(
kk
k
kk
kk
= 2
2 5
120 1
k
k kk k k
kk
k
!( )( )
= k k
k kk
( )
( )( )
3
1 21
J1061
= 12
11
4
2 122
1
k k kk k( ) ( )
J1061, Marsden p. 358
= dx
x
x
xdx
( ) ( )1 140
2
40
=
0
5
0
2
33 xx e
dxx
e
dxx
= sinh x
edxx2
0
1 .333333333333333333333 =
0
2
)2(4
1
2
1,2
3
4
kk k
k
k
=
122
11
kk
.333550707609087859998... log csclog ( )
2
3 2
2
2 1
21
k
kkk
.33357849190965680633... 1
4 11k
k
( )
.33373020883590495023... sinh sin1 1
...
6
12 2
( )
.33391221206618026333...
0
1
12
2log
8
1
312812
)1(
k
k
k
... ( )k
kk
1
2 11
... 3 2
4 16
2 2
21
log arctan
Gx
xdx
...
10
51
50
1
2520
coth
kk
.33432423314852208404...
84 4 2 4 1 6 2 arcsinh log \
= arctan arcsinx x0
1
dx
... 622
2
1
ek
k
k
k
!
... log( )k
kk
22
1
... e /2
... 2 2
... 11
21
kk !
... 41
41
2
1
40
1
7 221
tan
k kk
... e
ee e e
k
e kkk
11
10
log( )( )
... 6
720 volume of unit sphere in R12
...
( )2
21
kk
k
... 3
3
... log log ( )( )
2 2 12
11
2
kk
k
k
k
= 1
2 11
21 k kk
log
... 2
2e
= 42
125 6
2
1
kk
k
... 46
75
2 2
5
1
2 51
log
( )k kk
... 4 41
1 2
1
21
Gk
k
k
( )
( / )
...
22
2
)()1(
log
kk
kkk
k
.336409322543576932... ( )k
k kLi
kkk k
1
2
1 1
221 1
2
... Li12
7
... 2 3
3
2
120
12
3
2
32
1
2
log log log
( )
Lix
xdx
... e
e2 1
.33689914015761215217... 14
3 32 2
1
2
2
0
2
log( cos )
sin
/ x
xdx
... ( )
( )!/2 1
11
1
2 1 2
2
2k
kk e k
k
k
k
... Likk
k5 5
1
1
3
1
3
... 3
2 4 2 3 33 1
12
121
log log( )!
( )!( )
k
k kk
... ( )
( )!!
/2
2 21
1 2
21
2
k
k
e
kk
k
k
...
1 21)2(
k
k
kk
... 4
8 6 2 CFG D1
... k
k kk
1
2
... 6
25 5
1
2 1
1 12
12
1
( ) ( )!( )!
( )!
k
k
k k k
k
=( ) ( )!( )!
( )!
1
2 2 1
1 12
12
1
k
k
k k
k
...
2 2
3
3
21
3
2 22
sinkk
... )1(Re)2()!2(
)1()1(
1
Eikk
cik
k
AS 5.2.16
1 .337588512033449268883... 2
0
1
48 4
2
2
loglogarccot x x dx
... 2 11
2 11
log/
e
dx
ex 2
.33787706640934548356...
3
22
2 1
11
log( )
( )
k
k kk
... ( )
( )( ) log
2
11 1 1
1
1
22
22
k
k kk
k
kk k
= 2log2
1
... 2
3 21
3 216
2 21
logk k
dx
x xk
... ppf k
kk
( )
22
.33868264216941619188... 68 1
51
ee
x
dx
xcosh
... 1
1 20( ) !e
B k
kk
k
... 18
241
12
0
1
x
xx dxlog log
=log( ) logx x
xdx
13
1
... 12
2 3 14 3
1
22
2
2
32
( ) ( )
( )k k
k k
k kk k
1
...
2 2
8
3
2 1
2
0
2
x dx
ex
.339395839527289266398... ( )
log2 1
4
11
1
412
1
k
k jkk j
j
... 16
15
2
1 2
/
3... 761
2240
1
1625
kk
... e
8
... 2
31 2
1 3
21 2
1 3
2
2 31 3
1 32 3
1 3
//
//
/( ) ( )
i i
2
32 2
2 32 3
//
= 4 3 14
4321
k
kk
kk
( ) )
.339836909454121937096... arccsc 3
... ( )
!( )!
k
k k kI
k kk k
1
1
12
1 1
21
2
... 2
1
( )
!
k
kk
... 2
29
... 1
9
2
3
4
271
1
9
4
3
4 2
9 2
1
3 11
21 2
21
( ) ( ) ( )
( )
g
kk
= log x
xdx3
1 1
... ( )
!( )
k
k kk
1
12
.340791130856250752478... Likk
k4 4
1
1
3
1
3
.34084505690810466423...
1
2
1 1
1
1 12
1
( ) ( )! ( )
( )!
k k
k
k
k
1 .340916071192457653485... 1
21 F kkk
1 .340999646796787694775... 1
3
5 3
27
12
11
k
kk
.34110890264882949616... 27
1 2
1
1 12
1
( ) ( )!
( )!
k k
k
k
k
... 9
16
1
2
4
0e
k
k
k
kk
( )
!
... ( ) logk kk
1 11
... 5
2
15 2
1
kk
/
.34153172745006491290... 1
2
1 3
26 31
1
1k k
k 3
k
k
kk
( ) ( )
4 .341607527349605956178... 6
... 3
5
.34176364520545358529... 1
2
1 4
26 21
1
1k k
k 2
k
k
kk
( ) ( )
... 2
2
1122
2 1
log( )
H
k kk
kk
80 .34214769275067096371... 43 13
0
ek
k
k
k
( )
! GR 1.212
... log ( ) k
kk
2
2 .34253190155708315506... e e
... k k k
kkk k
( )
( )
3 1
2
2
2 11
4
3 21
.3426927443728117484... ( )4 1
4
1
415
1
k
k kkk k
=
1
41
1
21
21
1
21
2
i i
52 .34277778455352018115... 945
32
11
2
.343120541319968189938... 8 2 3 41
4 5 31
log ( )
k kk
2 .343145750507619804793... 8 4 2
... 3152
2796 3
1
2
1 40
1
( )
log/
x
xdx
... sin
( )
x
xdx
1 20
.3433876819728560451...
0
1
12
2log
4
1
312412
)1(
k
k
k
... H
k kk
k
2
1 2 2( )!( )
... 12 6 1 10 11
2 20
cos sin( )
( )!( )
k
k k k
.3436037994206676409... 1 1
10
e
e
e
k
e k k
k
( )
( )!
... 1 1
10
e
e
e
k
e k
k
( )
( )
k
!
1 .3436734331817690185... 1
2 311
kk
... 2 4
2
7
12012 2
9 3
2 log
( )
=
log log1
1
0
13
xx dx
16 .344223744208892043616... k k
kk
3
2
( )
!
... 2
3 2118
11
54
1
3
k kk
... H k
kk
( )3
1 4
2 .34468400875238710966... 1
4
1
2 10
e erfie e
erfi ek
k kk
cosh
!( )
.344684541646987370476... 265 72061
ek
k kk
!( )
.344740160430584717432... 2
12
2
20
2
cos
cos
x
xdx
5 .3447966605779755671...
csc5
1
5
4
5
= log 1 5
0
x dx
...
4 1 40
2
ex e x dxx
/ sin
.3451605044614433513... 9 4
11900001
2564
5
sinh
kk
.34544332266900905601... sin cos
( )( )
1 1
41
2
2
1 !
k
k
k
k
... 1
2 22 1 2
0
e ek
ke e
kk
/ / sinh
!
... ( ( ) ) log k kk
1 2
2
...
1
2 2 2 2
2 1
2
1
2 112
2
cot
( )k
kkk k
1 .345506031655163187271... 1
2 2 2 2
2
2
1
2 112
1
cot
( )k
kkk k
...
logcos
sin
arcsin/
2 21 10
2
0
1
G
x x
xdx
x
xdx
= K xdx
x( )
20
1
GR 6.142
.345729840840857670674... 1
32
1k
k
.3457458387231644802... 1
2
2
2
1
1
2 10
20
I
e k
k
k
k
k
( ) ( )
( )!
... 1
21
21
21
1eEi
H
kk k
kk
log ( )!
... 2
1
3
1
3 2 12 10
erfik kk
k
! ( )
... 2
0
1 x
xdx
... ( ) sin ( ) ( ) sin ( )/ / /1 1 1 13 4 1 4 1 4 3 4 /
... 5
5
2
1
5
1
5 521
tan
k kk
.34649473470180221335...
( )
( )
( ) log2
22 22
k k
kk
.34654697521433430672... 4
32
5
21
11
11
20F
k
k
k
k
, ,( )
( )
!
... logRe log( ) Re ( )
2
21 1 i Li i
= arctanh1
3
1
3 2 12 10
kk k( )
AS 4.5.64, J941
=
111 2
1
2
1
2 kk
kk
kE
LiH
= ( ) cos /
1
2 2
2
0 1
k
k kk
k
k
=
( )( ) log
2 1
21
1
21
1
1 12
2
k
kk
kk k k
= dx
x x
dx
x x( )( )
1 313
1
= x
x x xdx
1
1 120
1
( )( ) log GR 4.267.4
=
002
112xx
e
dxx
e
dx
=
4/
0
4/
0 sincos
sincostan
dxxx
xxdxx
= ( )cos
10
ex
xdxx
= GR 6.496.2 ( ) sin sinx x x30
1
dx
= x x
xdx
2
20
sinh
cosh
GR 3.527.14
.34700906595089537284... ( )
11
13
1
k
k
k
k
... 2 2 2 2 22 2
2
1
log log( )
H
kk
kk
... Likk
k
1 2
1
1
2 2/
.347296355333860697703... 218
2 2 2 2sin ...
[Ramanujan] Berndt Ch. 22
... 6 5
04 1 1
log
( )( )
x
x xdx GR 4.264.3
.347312547755034762174... 24 5 12 32
5 1
4 4
30
( ) ( )( )
x
ex
.347376444584916568018... 32 2131
6
1
2 50
log( )
kk k
.347403595868883758064... erf1
... ( )
logk
kk
k
1
2
.34760282551739384823... 4
3 3 2
1
2 2
2
20
2
cos
( sin )
/ x
xdx
...
2
32 2 1 2
1
2
3
2
1 31 3 1
2 3 2 3
// / /3
/ /( ) i
2
31 2
1
2
3
2
1 32
2 3 2 3
// /3
/ /( ) i
= ( ) ( )
1 2 3 12
21
13
2
k k
k k
kk
.3477110463168825593...
1
1)1()(k
kk
1 .348098986099992181542... 3
23
... 2
3 902 3 16 2 16
1 4
2
2 4 1
1
( ) log( ) ( )k
kk
k
= 1
2 5 41 k kk
1 .348383106634907167508... i i log
7 .3484692283495342946... 54 3 6
7 .348585884767866802762... 1
33
1
2
3
21
cot tank kk
Berndt ch. 31
... Likk
k3 3
1
1
3
1
3
... 9 1
7 2
90
x dx
x
/
.3497621315252674525... 42
3 21
4
1 1
421
1
1
log
( ) ( )
k k
k
k
k
kk
= hg1
4
5
4
= 1
0
4 )1log( dxx
... primepp 12
1
... coth21
21 1 2 1 2 i i
= 4 1
41 4 2 2 13
1
1
1
( )( ) ( ) ( )
k
k kk k
k
k
k
k
... 1
2
5
3
1
3
2
31
1
3
2
30 1
3
0
1
, , , e dxx
... k
kk
k2
2 11
( )
= 20
7
... 11
220
k
k
...
4
1
2
1
2 1
2
20
4
arctancos
cos
/ x
xdx
.35024515950787342130... | ( )| 2
4 11
kk
k
.35024690611433312967...
6 3
2
3
3
6 131
2
log log x dx
x
... 2 1 2 21
2 11
sinh( )
kk !
... ee kk
12 1 2
1
2 11
sinh( )!
dx
= GR 3.915.1e xxcos sin0
... E1
2
... 52 42 15
1
2
1
ek
k
k
kk k
!
( )
! Berndt 2.9.7
.350836042055995861835...
2 1 31 3 2 3
2 3 2 3
6
2
61 2 1
4 2 2 3
4
/
/ // /
( )i
( ) /
/ / /
12
6
4 2 2 3
41 3
1 3 2 3 2 3
i
= ( )( )
13 2
2
1
21
5 211
kk
kk
k
k k
1 .3510284515797971427...
( )( )
( )
3
2
3
42 1 1
4 3
12
2
4 2
2 32
k kk k
k kk k
1
.351176889972281431035... 2 2
24
2
8
log Berndt 9.8
15 .351214888072621297967... ( ) ( ) ( ) ( ) ( )2 6 4 6 3 1 13
2
k kk
= k k
kk
2
42
4 1
1
( )
.351240736552036319658...
4 5 4 520
dx
x
.351278729299871853151... 21 20
ek
k kk ( )!
... K Kdk
k2
0
12
2
( )k GR 6.143
.35130998899467984127... 1
2
1 4
251
1
1k k
k 1
k
k
kk
( ) ( )
... dxx
x
1
0
22
1
arcsin
2
3216
... 4 2 3 1 3 121
e e eEi EikH
kk
k
( ) ( )( )!
... ( )
logk
k kkkk
1 1
2
11
1
221 k
... e2
.3520561937363814016... 1
4 1 422
11 ( )
( )k k
kk k
k
... 2 31
4 2 1 2 30
arctanh1
2 kk k k( )( )
... 1
3
1
3
1
3 2 11
sinh( )
kk k !
... 22
11 2e isiin ei ei i
( ) / cos cos ( )i
... 4
4
... 1
33
12
13SinhIntegral
x
dx
x( ) sinh
... 2
28
... arccot e ee k
k
kk
2
1
2 12 10
arctan( )
( )
= dx
e ex x
1
... 1
22 2 1
2
12
20
e Eik
k k
k
k
( ) log!( )
...
1
02)1(
)2()1(2
11
xe
dxEiEie
e x
... 54 193
3
1
ek
k kk !( )
... Jk k
k
k
k
k
k
k2
0
3
20
21
21( )
( )
!( )!( )
( !)
LY 6.117
2 .35289835619154339918... ( )k
k kLi
kk k
1 12
12
1
1
1 .352904042138922739395...
22
3172
5 4
4
1
25 4 2
( )( ) ( )
H
kk
k
Berndt 9.9.5
.3529411764705882 = 6
17
.35323671854995984544... 1 1 11
2
p k
kp prime
probability that two large integer matrices have relatively prime determinants Vardi 174
.35326483790071991429... 2
1 100
1
( ( ) cos ) sin arcsinJ x x dx
.3533316443238137931... ( )
1
2 1
1 2
1
k
kk
k
... x dxx1
0
1
/ 1 .353540950076501665721... ( ) ( )3 2
.353546048172969039851... cos sin sin1 1
2 1e e
x x
edxx
... 8
sin8
3sin
8
8
4
2
...
4
1
2 6 2
12
4 21
4 21
cothk k
dx
x xk
...
1 5
)2(
5csc
5log
kk k
k
... log( ) ( )!( )
e Lie
Lie
B
k kk
k
1 1 21
21
2 322 3
0
= x dx
ex
2
0
1
1
... 2
3
.35417288226859797042... ( )4
4
1
4 114
1
k
kkk k
= 1
2 4 2 2 2
cot coth
... ( )( )
!
12
k
k
k
k k
.354248688935409409498... 3 7
.35429350648514965503... 1
431 kk
.35496472955084993189... 11 1
2
2 1
40
1
1
e
Ik
k k
k
kk
( )
!
... 2
2 12
1 2 0
1
log arctan arccosx x dx
.355065933151773563528... 2 21
1
4 1
2 2 121
2 21
( )( ) ( )k k
k
k kk k
=
212
23)1()()1(1)2(
1
k
k
kk
kkkkk
= ( ) ( )( ) ( )
1 21 1
21
1 2
k
k
k
kk
k kk k
= log GR 4.221.1 log( )x x10
1
dx
.355137311061467219091... 7
1920
4 3
31
log x
x xdx
4 .355172180607204261001... 2 2 log
= x dx
ex
10
GR 3.452.1
=
02
2
1
)9log(dx
x
x
= x x d
x
sin
cos10 x
GR 3.791.10
=
2
0 2sinlog dx
x
= GR 4.224.12 log ( cos )2
0
1 x dx
= GR 4.226.1 log (cos )/
2 2
0
2
x dx
... 5 3 1
6 2 1 12 50
dx
x /
.355270114150599825857... 3
2
2 1
11
11
1
22
2
log( )
log k
kk
kk k
.35527011687405802983... ( ) ( ) tanh3 2
3
3
2
15 4
1
3
k k kk
.3553358665526914273...
1
2
)!2(
)2()!(
k k
kk
.355379722375070811280... 1
3
2
3
3
2
1 4
32 2
1 30
2 3 1 3
e ek
k k
k
/ /
cos( )
( )!/
... )3(
)6()3(
.35577115871451113665... 1
1342 kk
1 .35590967386347938035... 11
41
kk
... coth kk
11
... 3
41
22
1
arccot ( ) arctankk
K Ex. 102
[Ramanujan] Berndt Ch. 2
=dx
x x20 2 2
... log7 2
... 4 2
1
21
2
4
1
4 11
211
coth ( )( )
kk
kk
k
k
.356395048612456272216... 1
42 2
3
2
1
8 22
2 11
Fkk
kkk
, , ,
... ( )k
kk 41
.3565870639063299275...
( )
( ) ( )
( )12
23
16 1
11 2
3
k
k
J1028
.356870185443643475892... 4
3
1
4 4
1
3
1
4 12
2
12
0
LiH
kk
kk
kk
( )
( )
= 5
14
...
0 12
1
3
123xe
dxx
1 .3574072890240502108... kk
k
2
1 3 1
... 3 3
224 2
27 3
2
log
log
=1
1 12
130 ( )( )( )k k kk
.357907384065669296994... 1 11
2
1
21
cot tank kk
Berndt ch. 31
= csc1
22
0k
k
k
Berndt ch. 31
... 36195
2 3
4
0
ek
kk
( )!
... 2
21
1
2 21
sin
kk
1 .358212161001078455012...
0
2 )tan2sin(1sinh
12 x
dxx
ee
=
0
2/
0
cot)tan2sin(cos)tan2sin(
dxxxx
dxxx
1439 .358491362377469674739... 61
128
7 6
0
x
xdx
cosh GR 3.523.9
4 .358830714150528961342... 3 32
( )k
k
... 2
1
2
8
( )kk
k
... 19
...
13 14k k
k
...
00 xe
dxx
xe
dxxx
.359140914229522617680... e k
k kk21
2 41
!( )
1 .359140914229522617680... e k
k
k k
kk k2 2 1
1
21 1
( )!
( )
!
= dxex x
0
1 2
=
1
1 30 e x
dxx ( )
= log
( log )
x
xdx
e
1 21 GR 4.212.7
.359201008345368281651... 3 3 3 4 ( ) ( ) ... 3 4/
3 .35988566624317755317...
1
1
k kF
... ( )
!/k
ke
kk
kk
11
11
22
... arccot ( cos )cscsin
2 2 22
21
k
kkk
=
1
31
2)1(
25
34
kk
kk kF
.360164879994378648165...
1
16
1
2
3
2
1
16
1
2
3
2
i i i i i
i
= ( )( )
14 2
4 41
2
41
kk 1k k
k k
k
1 .360349523175663387946...
12 9/1)12(
1
4
3
k k
GR 1.421
... 11
81
k kk ( )
.360553929977493959557... 1 2 3 4 ( ) ( ) ( )
...
0 )2(2!
)1(64
kk
k
kke
...
12 64
1
4
32coth
22 k kk
... 1
4 2 22
1
2 321
coth
k kk
=
12 45
1
36
13
k kk
= 3)2(
36
49H
...
e Eidx
e xx2
0
22
( )( )
... 2
1arcsin
...
1
32 84
1
1621
coth
kk
J124
.361594731322498428488...
1
2
)1(32
3log
2
3log
2
3
kk
k
k
kH
5 .3616211867937175442...
3 2 2 2 2 3
2 4
6 2
4
3 2
4
2
23
( ) log log log( )
=
log3
20
xdx
e x
441 .361656210365328128367... 162 11
7
1
ek
kk
( )!
...
0
3
)!1(
3
3
1
k
k
k
e
... 1
12 2 2 1 2
42 3 2 3 1 3 2 3
3
32
( ) ( )/ / / /
k
kk
= )4)(1(4
12
105
38
0
kkk
kk
k
.3620074223642897908...
1 )12(
sin
k k
k
... 275
6 3
1
1 3
3 1
31
( )
( /
k
k k )
... ( )( )
( )
12
2
1
4
112
12 2
1
k
k k
k
kLi
k
... Ik k
k0 2
1
2
31
1
3
( !)
...
1 )12(2
11
kk
...
11
1
)(
))((
)(
)1(
kk
k
kp
kp
kp
, p(k) = product of the first k primes
.362360655593222140672... 1
22k
k k( )
... 1
31 ( )k kk
.3625370065384367141... 29
15 21
4
2 1
1612 1
1
( )
( ) ( )kk
kk
k
k k
... erfk k
k
kk
1
3
2 1
3 2 12 10
( )
! ( )
... 35
.3632479734471902766... 1
23 3
2
22 2
0
1
(log loglog( )
x
xdx GR 4.791.6
... 4
3
3
2
.363297732806006305...
sin( ) sinh( )
3 3
241
92 4
2 k
...
1
01 log
1log
)1log(
x
dx
xe
e
ke
k
kk
GR 4.221.3
... 12 2 1
2
1
2 1
2
1
( )!!
( )!!
k
k kk
J385
= ( )( )!!
( )!!( )
12 1
24 11
3
1
k
k
k
kk
.36363636363636363636 = 4
11
... 1
12 k kk ! !
... 1
21 Fkk
... 2
23
1
7 3
41 1
2
4
2 1
( )( ) ( )
( )
( )k
kk k
k kk k
k
...
1
33/1
3!1(9
173
kkk
ke
...
234
234
1
2
32log)2()3(
xx
dx
kkk
.363985472508933418525...
01
2
1
1
sin
1
)1(
sinh22
1x
k
k
e
x
k
sin
cosh
2
20
x
xdx
...
12 26
17cot
7214
1
k kk
...
1 )!1(
11
k k
...
0 )12(!
22
22
1
k
k
kkerfi
...
1 )!2(
42log)2(2
k
k
kkalCoshIntegr
.36481857726926091499... 12
2 21 21
1
log( ) ( )
k
k
k
.36491612259500035018... 3
3 6 7 CFG A22
... ee xx
4
2 2 1
0
2
log log dx
...
0
2 4/1
)1(
2csch2
k
k
k
.3651990418090313606... 31
3 1
1
3
1
332
1
1
( )( )
kk
=( ) ( )
1 1
3
1
1
k
kk
k k
1 .365272118625441551877...
1 1
k
k
2 .365272118625441551877... 1
1
.365540903744050319216...
2
31
2
31
3
1
27 22
2 iLi
iLi
64
9
8
27
64
272
31 2
22
0
log ( )/x Li x dx 26 .36557382045216025321...
.365746230813041821667...
22
11log
11
1
)1(
1)(
kk kkkkk
k
...
22 22
11log
11
11
)(
)1(
)(
kk jj
k kkkk
k
kk
k
22 1
2log
k kk
k
...
1 .365958772478076070095...
11
1 11
2)12()!1(
)2()1(
kk
k
kerf
kkk
k
13
3
...
= cos sin 6 6
1 12
= 6 31
( )k
k k J1029
.36602661434399298363...
( )3 1
2 13 11
kk
k
1
4
1 1
1 1
3
2 962 1
3 4
3 4 23 4
0
4
log( )
( )( ) sec
/
//
/
G i
iLi x x dx
...
log 3
3 41
H Ok
kk
...
036/13
3/1
63
2
x
dx ...
3
23log3log2log)2(
3
12
22 LiLi .36621322 99770634876...
= Lik
H
kkk
kk
kk
2 21
1
1
1
3
1
3
1
2
( )
.36633734902506315584... 3 22 4
7
9
8 1
4 1
2
22
log( )
Gk
k kk
= ( ) ( )k k
kk
1 1
41
1
log log 2
4449934445032...
...
1
1296
1
3
5
6 13 3
3
31
( ) ( ) log
x dx
x .36656347
2 1 1 1 2 1371
97202 3
42 3
41 3
42 3
4
( ) ( ) ( ) ( )/ / / / Li Li Li
=
70 8 .3666002653407554798...
... 1160703963
8388608 9
9
1
kk
k
( ) ( )3 2 4 ...
22
0
1
16
1
4 x xarcsin dx ...
00 cosh12
)1(2
4
3,
2
1
4
1,
2
1
xx
dx
k
k
...
4
G ...
134
1
12
2
1
2
)3()1()3(4log4
38
kkk
k
kk
k ...
.367042715537311626967... cos ( )( ) ( )
2
31
32
323
2
1
4
k
kLi e Li e
k
i i
.367156981956213712584... ( )( )
12 1
1
1
kk
k
k
k
k ks d
k
1
132 2
3
lo ... s( )
g
1 .367630801985022350791... ( )k
kk 21
1 1
22
0 0ei
k ki
k
k
k
k
/ ( )
!
( )
( ) ...
1
!
= ( )
!
( )
!
1 13
0
4
0
k
k
k
k
k
k
k
k
1
2 1 2 30 ( )!( )k kk
=
= ( )k
ekk
11
= k k e
k e kk
( )
( )(
1 11 )
J1061
dxex
x
e
dxx
02
1 )(
log =
= 3
1
1
0
1sinhsinh
x
dx
xdxxx
( ) log ( )
11
k
k
k ...
sinh
( )! (
2
2 2 2
2
2 1
2
2
2 2
0 1
)!
1 .36829887 20085906790...
e e
k
k
k
k k
GR1.411.2 k k
=
122
21
k k
... 2k
2
)1( k
kk
= 7
19
( )
( )
( )2
3 31
k
kk
... Titchmarsh 1.2.12
...
)1()1(2
1)(Re )2()2()2( iii
=
0
2
133 1
cos
)(
1
)(
1dx
e
xx
ikik xk
.368608550929364974778... 2 2
2116
2
2
2
16
7 3
16 4 2
log log ( )
( )
H
kk
k
1 3!
1
k
k
kk
...
2
1 100
1
( ( )) sin arccos J x x dx .3688266114776901201...
02 3/1
1
3coth
2
3
2
3
k k
...
.36966929 9246093688523... 12
2
21
1
21
1
log( )
( )k
k
k
k
= 1
21
11
1 kk
kk
log
3 3
4
33
2
31
4
9 622
2
log ( )k
k k
kk k
...
0 2 ),2(
!
k kkS
k ...
6 2 1 1
2
120
8
120
x dx
x
x dx
x ...
363
980 7
1
77
21
H
k kk
...
( )
( )
3
3 1
1
313 3
1
kk
k kk .3704211756339267985...
dxx
x
1
03
6 )1log(
344
3log3 .3705093754425278059...
2 .370579559337007635161...
2/
0
224
cot)tan2(sin1
34
dxxx
e GR 3.716.11
2
1
k
kk k
...
5
21 1
1
4 120
log( )x
dx .3708147119305699581...
1 )!2(
33sinh
2
3
k
k
k
k ...
4 1
220
1Gx x
sec ... dx
55290132347...
1
22
2
1)2(2
2csc
82
2sin1
kk
kk .37133401
=
222
2
)12(
2
k k
k
2 .3713340155290132347... 12
2 8 2
2
2
22
1
sincsc
( ) k kk
k
= 2
2 1
2
2 21
k
kk ( )
.37156907 160131848243... G( H
k
kk
k
log ) /2
4
1
2 1
11 2
0
Adamchik (25)
=
12 )12(4
)2()14(
kk
k
k
k
= dxx
x
1
021
)2log( GR 4.295.13
= dxx
x
1
02
2
1
)1log( GR 4.295.13
=
0 cosh
)2/cosh(logdx
x
x GR 4.375.1
.3716423345722667425...
1 2 215 2
6
2 2
3
3
2
2 3
( ) log
log log ( )
= log ( )
( )
2
2 20
1 1
1
x
x x
1
44 2 1 4 1 22 (cos ) log( sin ) ( ) sin ... 2
=cos
( )
2
1 1
k
k kk
1
1
2)1(
2arccsch55
4
5
1
k
k
k
k ...
87240681867... 1
4 121 k kk !( )
.37218495
( ) sin logcsc log sin 2 1 1 1 22 ... 2
=
sin( )
( )
2 2
11
k
k kk
GR 1.444.1
1
2
1 3
241
1
1k k
k .37252473265828081261...
1
k
k
kk
( ) ( )
0 )1(8
12221624
kk kk
k ...
... 1
41 12 2 1 1 coth ( ) ( )( ) ( ) csch i i i i
= ( ) ( ) ( )( )
1
12 2k
( )
1 2 211
1 2
k
k k
k k kk k
...
0
22
3
12log2
3 xe
dxx GR 3.452.2
.3731854421538599476... 1
4 1 41
11 ( )
( )k k
kk
k
k
...
.37350544572552464986... 1
2 1
1 4
241
1
1k
k
k
k
kk
( ) ( )
16
1
46
1
3
2log
33 1
kk J80
k
...
= ( )
1
3 20
k
k k K ex. 113
= dx
x
x dx
x
dx
e ex x31
30
1
201 1
.37382974072903473044... 20
9
8 2
3
2 1
4 2 20
log
( )
k
k kkk
( ) ( )log3
3
31
3k
kk
...
2/
0
22
cos12log
44
dxx
xG ... GR 3.791.5
...
4 2 2
61
122
0
1
loglogx
xdx
14
2 )1log(dx
x
x =
( )
( )
1
2 2 1
1
1
k
k kk
k ...
0 )4(2!
15896
kk kk
e ...
02
)1(
)(
1)(
k k ...
1
1
1 1
23 20
2 2 3 122k k k k k k k
1
k kk
1 .37427001649629550601...
ii i i i
1
41( ) ( ) ( ) =
1
2 4 2
1
4
coth ( ) ( )i i =
( )k
kk
1 1
121
...
12 e
...
cos log1 2 ...
1
22k
k k ( )
...
0
/1
!
11
kkk
e
...
/1e ...
...
1227
222 xx
dx
xx
dx
... ii ee 222
11
1)1()cos()1(1
1
kkk
k =
1 2 3 13
01
21
Jk
kk
k
( ) ( ... )( !)
00000000000003
82 1 3
0
e k
k
( ) log .37500000 =
= )4
1 2(
1
k kk
=
13
3log
x
dxx
2 2 2
404 24
2 2
22
log
loglog xdx
e x 1 .3750764222837193865...
dxx
220
1
.3751136755745319894...
ii eEieEi2
1 ...
1 !
cos,0,0
2
1
k
ii
kk
kee =
4
105
1
1
2 2
2 4
p
p pp prime
...
1
1
1
2 21k kkk
j kkj
...
Likk
k
1 2
1
1
4 4/ ...
4
2log2
G ...
5cot
...
2 1 2 2 21 2 1
2
1
1
log log( ) ( )!!
( )!
k
k
k
k k .37645281 29191954316...
1
1 2
4
)1(
kk
k
k
k
k =
12 54
1
5
6coth
2 k kk
...
0 )3()!12(
148
659
k kkee ...
11
2 2 2 2 11
arctanh1 k
kkk ( )
...
=( ) ( )!!
( )!!
1 2
2 1
1
1
k
k
k
k
1
22 2
0
1
G x x xlog ( ) sin GR 6.468 d 1 .37692133029328224931...
13
2
)2(
1
16
3
242
)3(
k kk
...
1
1
1
2
1
2
1
41 1 2ek k ( ) /
1e k
tanh ...
1
2
1
2 1
1
221
( )!k
B kk
= J134
1 .37767936423331146049... 2
33 4 1 2
22
2
( ) ( ) ( ) ( )k
k
k k k
= 4 2
1
3 2
3 22
k k k
k kk
( )
1
...
22
log1
k k
k
7
7 ...
ex x
e(cos sin )log coslog
1 1 3
23
1
... dx
... e
x dxe(cos sin )coslog
1 1 1
2 1
.378139567567342472088... 2 42
3
1
2 20
log( )
( )
k
kk k
12 35
1
2
13tanh
131
k kk
...
dxx
xHypPFQ
13
sin
44
1,
2
3,
2
1,
2
1 ...
dxx
xsici
02)1(
cos1cos)1(1sin)1(
2
1cos1
...
.378605489832130816595...
...
( ) ( ) ( )
1 11
2
k
k
k k
...2
1
2
1
2
1
1 212
12
kkkk kk
0
/1
!
cosh)1(
2
1
k
kee
k
kee ...
2 23
22
2 1
2 1 22 2
1
log log
H kk
k
...
3
2 32
1
5 30
log
( )
/
k
k k ...
3 3
22 6 2
42
23 23 2
2 2
( )log log log 4
x
...
= a rcsin logx x d3
0
1
2
26 ...
( ) ( )
( !),
12
11
12
20
1
k
k k
k
kJ
k k
...
4/
0
6
sin1
cos
20
1
40
2
32
3 dx
x
x ...
log( ) ( ) ( )
!
2
1
1 1 2 1
1
e
e
k
k
k k
j
...
dx
ex
10
=
.37989752371128314827... 3
3
0
1
32
3
16 x xarcs dx in
... ( ) ( )
1 12
2
k
k
k
( ) ( )
1 12
2
k
k
k ...
.38079707797788244406... e
e
B
k
k kk
k
2
2
2 1 22
1
1
2 1
2 2 1
2
( )
( )
( )!
88 2 22 9 .3808315196468591091...
2
53
10
55
2
53
10
55
2
5tan
51 ...
=
2
22 )1(
11)()1(
kkk
k
kkk
kkF
0 1
sindx
e
xxx
...
1 .381713265354352115152... 1
21 ( !!)kk
cos sin ( )( )
( )!1 1 1
2
2
2
1
k
k
k
k ...
log ( )( )
12
12
12
41
1
i i
kk
kk
k ...
= log 11
4 21
kk
.381966011250105151795... 23 5
2
1 2
121
( ) ( )!
( !) ( )
k
k
k
k k
2 .381966011250105151795...
0 2
1134
k kF GKP p. 302
.382232359785690548677...
1 54
5log1
16
5
kk
kkH
( )
1
2 22
k
kk
...
( )
!/3 1
11
1
1
3k
kk e
k
k
k
...
12
1 15
1
5
)2(
5cot
522
1
kkk k
k ...
.38259785823210634567... dxx
x
14
arcsinh
6
)21log(2
( )3
...
.38268343236508977172... 8
3cos
8sin
4
22
.382843018043786287416... ( )( )
14 1
1
1
kk
k
k
... log
( ) ( )2
2
1
8
1 2
2
1 2
2
i ii i
= cos2
0 1
x
edxx
log
1
1
1 k kk
...
... 1
.3834737192747465...
( ) ( )
1 2 11 2
1
k
k
k
21 !
log
2!
1
k
n
nn k
k
n
.38349878283711983503... 8 2 4 2 161
0
1
Gx x
x
log
log( ) log
.38350690717842253363... ( ) ( )2 2 11
k kk
1
.383576096602374569919...
1 43
4log
3
4
kkkH
= 7
18
.383917497444851630134...
12 16
122cot
24112
61
k kk
.383946831127345788643... 2
2
1
31
1
1
2
( ) ( )
( )
k
k kk
.38396378122983515859...
23 1k
k
k
H
... 11
2
2
2 11
21
2 1
2 11 1
kk
k
kk
kk
Q k( )
.384255520734072782182...
12 2
11
k k
.384373946213432915603... dxxx 1
0
)(log)2(sin2
2log
GR 6.443.1
.3843772320348969041... ( )4 1
4 414
1
k k
kk 1k k
= 1
8 2 2
1
2
1
2
i i
= 1
22 1
1
21
1
22
1
22
1
21HypPFQ , , , , ,
1
62 1
21
22
22
21HypPFQ
i i i i, , , , ,
.384418702702027416264...
0 )3)(1(3
1
3
2log12
4
21
kk kk
1 .384458393024340358836... dxxx
x
1
021
arctan)21log(
2
GR 4.531.12
... 1
3
2
3 6 12 62
2 2
40
4
G x
xdx
logcos
/
1 .384569170237742574585...
0
22
)1)(3(
log
8
3log
8dx
xx
x GR 4.232.3
... sin cos log( cos ) ( ) sin1
2
1
22 2 2 1 1
1
22
= sin( )
( )
k
k kk
1
11
GR 1.444.1
=5
13
.384654440535487702221... 3
2log
3
2
2
3log3log
2
2log3log2log)3( 2
322
Li
3
2
3
133 LiLi
=
1 1
)2(1
2 2
)1(
3k kk
kk
kk
k
H
k
H
7 .38483656489729528833...
3
2)2(
.38494647276779467738... log log2
4 2 220
x
x
.3849484951257119020... ek k k kk k
7
3
1
3
1
1 21 1!( )! ( )! !
... ( ) ( )/ / / / / /
1
1
44 2 2 3 1
1
44 2 2 32 3 1 3 1 3 1 3 1 3 1 3 i i
1
3 21
1
22 3 2 3/ /
= ( )3
4
1
4 1311
k
kkkk
.38499331087225178096... )1(!2
1)1(2
2
1
0
11
kkJ
kk
k
.385061651534227425527...
22 1
1
2
5arctan
3
2
3 xx
... 3
13
... 6
...
3
1
61 3 1 3
1
331
i ik kk
5 .38516480713450403125... 29
.38533394536693817834... log cos ( )cos
2 1 1 12
1
k
k
k
k
... 1)1()sin()1(222 1
1
kkeei
k
kii
... /1
... arccothee kk
k
1
2
1
2
1
2 12 10
log tanh( )
J945
= dx
e ex x
1
...
12
1
1 1
2)12()!1(
)24()1(
kk
k
kerf
kk
k
... 1
21
2
0
e xx ( )dx
... 2 2 11
2 1
1
2 2 401
log( ) (
k kkk k k )
= 1
4 121 ( )k kk
J374, K Ex. 110d
= ( )
( )
1
1
1
1
k
k k k J235, J616
= ( ) ( )
( )( )k k
kkk
kk
kk
1 1
2
2 1
41
1 21
k
= 2
1
11log
x
dx
x
= E kdk
k( )
10
1
GR 6.150.2
... 2 2 43
4
3
411
log log
Lik
k
kk
= ( )
( )
k
k k k
Hk
k kk
k
kk
k
1
2
1
2
1
2 1 212
1 0
1
=12 1
4 1
2
2 21
k
k kk
( ) J398
= 1 21
8 231
k kk
[Ramanujan] Berndt Ch. 2, 0.1
=
11 4
12
)!2(
!)!12(
kk
k kk
k
kk
k
=Hk
kk 21
= x x x
xdx
n n n
1 1 2 2 1
0
1 2
1
/
GR 3.272.1
=log( )1
21
x
xdx
= log 11
0
1
x
dx
= log( )
11
20
x xdx
...
1 2
2log22k
kkkH
... 27 2 22 1
4
1
log( )
k
kkk
... 925 2 22 1
6
1
log( )
k
kkk
.3863044024175697139... k
kk 4 11
...
12
1
kke
...
4 4 2
1
2
22
32coth coth csch csch
2
= ( ) ( ) ( )
1 2 21 2
1
k
k
k k k
... 1
4 4 5
5
2
1
4 421
tan
k kk 4
... log6 2
... Lie kk
k3 3
1
1 1
3
... G x
xdx
2
7
83
2
0
1
( )arctan
.38716135743706271230...
1
2
310
4)!(8
)1(
4
)1(
kkk
kII
.38779290180460559878...
125
1
5
21
24
5log5
55
55log
4
5
k kk
11 .38796388003128288749... 2
34 3
22
22
0
( ) ( )x Li x dx
...
1 )!2(
!!1
k k
kk
... 2
1arcsinh21log
2
1 22
=22
1
!)!12(
!)!2()1(
0
kk
k
k
k J143
= arcsin x
xdx
1 20
1
=
1
021
arcsinhdx
x
x
... 34
15 CFG D1
... ek
k
k
k
k
k
k
2
0 0
12 1
3
2 1
!
( )
! LY 6.40
... 6
...
123
1
2
4
1
4
)2(
622log12
kkk kk
k
.39027434400620220856...
1 )13(
11
k kk
.39053390439339657395...
0
12 )12(3!
12
3
1
kk kk
erfi
2 .39063... 1
22 ( )kk
...
0 1
2
),2(
),2(
k kkS
kkS
2 .39074611467649675415...
0 2
)3(222log8
kk
k
1 .39088575503593145117... log
2
3
.3910062461378671068... 64
22
0
1
arcsin arccosx x dx
...
12 544
1tanh
8 k kk
GR 1.422.2, J949
... 2 3 2
0
2
3
2
3 1
log log log
( )(
x
x x 2)
...
1
11
keke kee
Li
.39173373121460048563... sinh
cosh1
3
13 4
1
x
dx
x
.39177... 1
122
k s
ks
.391809...
2 )1(
)(
k kk
k
.391892330690243774136...
12
22
2
)1(22log2log4
62
kk k
k
... 2
2
04 2sech
x x
xdx
cos
sinh
.39207289814597337134... 16 2
22
1
( )
( )
... 4
2/3
1 .392096030269083389575... F kkk
2 12
2 1 1
( )
... 33
... e
e x x dxx4
1
0
2
log
.3925986596400406773... = 2 1 11
2 11
erfi Eik k kk
( ) ( )! ( )
...
1
4
!
44
k
k
k
ke
.392699081698724154808...
0 24
)1(
8 k
k
k
=
0 )34)(14(
1
k kk
=
1 02413
1
04
04 )1(44 x
dxx
xx
dx
x
xdx
x
dx
=
022 xx ee
dx
= dxx
xx
0
2 2sinsin
=
05
222 cossindx
x
xxx
= GR 4.523.2 1
0
arcsin dxxx
.3926990816987241548... = 12arctan8
= ( )
( )sin
1
2 1
2 1
2
1
21
k
k k
k J523
= ( ) ( )4 1 2
16
1
4 1
1
16 112
12
1
k
kk k k
k
k k
= 1
4 2 1 2 3
2
0k
k k k
k
k( )( )
= arctan( )
1
1 2 21
kk
[Ramanujan] Berndt Ch. 2
=1
4 2 211 ( )s r
sr
J1122
= dx
x20 16
=
0
coslog dxxxex x
= x x dxarcsin0
1
4 .39283843336600648478...
1
4
8 )()4()4(4
2160 k k
krL
.393096190540936400082...
12
22
0 )!(
2)1()22(2
k
k
k
kJ
...
12
2
)12(4
2log
4
)3(7
k
k
k
H
... 11
71
k kk ( )
.393469340287366576396... e
e k
k
kk
1 1
2
1
1
( )
!
... 2
6
...
3 !
1
k kk
... 2 2 1 2
22
2 22
k
k k
k
kLi
j
( )
... )3(34
...
0 !
8)1(50
k
kk
ke
... coth
8
... 3log4
...
1
)3(1
2
)1(
kk
kk
k
H
... 1)2(14
3log
1
kk
k
k
.394784176043574344753... 25
2
2 .394833099273404716523...
01 )!1(!
222
2
1
k
k
kkI
.394934066848226436472...
3
2)1(
2 1)3,2(3
4
5
6 k k
= 11
)(2
122
i
eLieLi
= ( ) ( )( ( ) )
1 12
k 1k
k k
=
2 11
1)1(
j kk
k
j
k Berndt 5.8.4
=
134
logdx
xx
x
= dxx
xx
1
0
2
1
log
1 .39510551694656703221...
3
)(k
k
... 11
2
kk !
...
( )
( )cos/1
1 20
e
ee xx dx
... ( ) ( ) ( )( )( )1 1
1
5
8
7
8
3 1 1
82
k
kk
kk
...
1 )5(!44120
k kk
ke
...
0
3/1
3!
1
kkk
e
...
1
22
)!3(
2
8
19
8
3
k
k
k
ke
... 2/7)2(
2
311025
51664H
...
1
21
!
1)1()1(
k
k
k
k
1 .39689069308728222107... k
k
kk
2log
1
12
.39691199727321671968... 2 2 1 12
2 11
1
sin ( )( )!( )
kk
k
k
k k
.39710172090497736873... 1
2 3
2
31
1
3 121
csc sin
kk
J1064
.397117694981577169693...
eerf
1
...
12
1
0 )!(
4)1()4(1
k
kk
kJ
.39728466206792444388... 3 3
4 2
3 2
20
sin x
xdx
... dxe
xx
e x
1
2 sin1cos2
...
0 )42)(12(
1
3
2log
6
1
k kk
3 .397852285573806544200...
1
2
)!22(4
5
k k
ke
.3980506524443333859... | ( )|( ) k k
kk
1
2
1
1
... 4
3 8
... 1
0
)tanarctan(arc dxx
...
2 6
2
3
1
4
1
3 221
coth
kk
... 1
2
... 13
2 ( )
.3990209885941838469... 2
2 2 2 2 21
2
20
cos cos ( ) sin( ) ( )sin
( )
si cix
x
... 1
4 17
17
2
1
5 221
tan
k kk
... e
k k k
k
k
2
0
1
16
2
2 4
!( )( )
... ( )
12
1
1
k k
k
Hk
kk
= 2
51
21
21
1
1
1
( ) ( )k kk
k
k kk
k
F kF
= 1
2 21 2 2 1
0cosh log( ) ( log )( )
k
k
e k J943
= sin sin cos2
0
2
02
0
x
edx
x
edx
x
edxx x
x
= 12
5 42
1
F kkk
k
... 121
1)1(
kk
Hk
... log 2
... e G
... e Eik
k k k kk k
1 11
1
221
20
( )!( ) !( )
= e x x dxx log0
1
... ( )( )
arctan
12 1
1 1 1
2 1
k
k k
k
k k k k
...
1
)2(log)(k
kk
.400685634386531428467... ( ) cos( / )3
3
33
1
k
kk
... 4 2
12 2
4 22 2
cos sincosh sinh
=sin2
40 1
x dx
x
... 1
62 1
1 22
( )
( )
( )(
k
k kk )
Admchick-Srivastava 2.22
... ( ) ( )
!
//k k
k
e
ke
k
kk
k
11
2
11
1
.40148051389327864275... log( )
27
24
1
4
1
23
k
k
k
k
... 912 1
3 21 30
e k k
k
kk
/
( )
! ( )
.40175565098878960367... 5 2
12
3
16 1
5
0
4
cos
sin
/ x
xdx
1 .40176256381563326007... log
1
3
...
6
1
3
1
6
1
6
53
4
=
06
1
03 11 x
dxx
x
dx Seaborn p. 168
1 .402203300817018964126... 3
46
23
0
23
0
1
x x dx x dsin arccos/
x
... 3
4
2
...
3
4
2
42
1
222
csc
( )k
k k
.40268396295210902116... Li32 34 3
5
4
52
2
3
( )( ) log log
5 1
2
=5
)3(4
2
15log
15
2
2
15log
3
2 22
... k
k kk
!
11
... ( )
!!
k
kk
1
2
.40306652538538174458...
03 2739
2
x
dxx
... e erfk k
k
1 18
0
12
1
3 2
1
3/
!!
6 .40312423743284868649... 41
... 0 1 0 20
11 2 1
Fe
Ie k ek
k
; ;( !)
... 2
1
2
2
( )
( )!
k
kk
... 1
8
1
3 13
2 230
1
log
( )
x
x xdx GR 4.244.1
... 98
2431
21
6
1
( )kk
k
k
... ( )
1 1
3 31
k
k k k
... K x
xdx0
2
20
2 2
2 1
( ) log cos
...
0 02 )1()1(
)1(1xe
dxx
xe
dxEie
xx
... 2 11
3
0
eEix dx
e xx( )( )
.403936827882178320576... 1
2 121
k
k
... 1
3 1
1
3 11
1
1k
k
k
kk
( )
... 2 3 2 22 ( ) ( )( )
= log log
( )
2
3 21
2
0
1 2
01 1
x
x xdx
x x
xdx
x
e ex x
GR 4.26.12
2 .4041138063191885708...
1
2)2( )1()3(2
k
k
k
H
=log ( ) log2
0
1 2
0
2
21
1
1
x
xdx
x dx
e
x
x xdxx
=log2
0
1
1
x
x
= dydxxy
xy
1
0
1
0 1
)log(
... 2 3 19 2
1
4 2
2 32
( )( )
k k
k kk
1
= (( ) ( ) ( )2 1 2 2 12
1
k k kk
... 82
3 1 1
41
2
1
Gk
kk
kk
( )( )
( )
...
2
31
6
3
10
cothsin
x
edxx
... 2 2 1
2 1
1 22 1
1 2
k
k k
k
k k
( )
( )!sinh
k
.4045416129644767448...
2 )1(
1)()1(
k
k
k
k
.40455692812372438043...
csc ( )
2
1
2
1
1
1
21
k
k k
... k
k kk
2
1
... 11 1
1
Eie k ek
k ! k
... 9 182
23
63
6 iLi e Li ei i( )
=
sin63
1
k
kk
GR 1.443.5
...
2
1
21 1 1coth i i
= )12()2()1(1
1
1
13
kkkk
k
k
k
k
... cosh sinh
( )
2
2
2
2 2
2
2 11
k
k
k
k !
.40496447288461983657... ( )2
... cos(sin ) cosh(cos ) sinh(cos )cos
!2 2 2
2
0
k
kk
... ( )
11
12
51
k
k
k
k
... 3 1
4
1
2 1
1 2
1
cos ( )
( )
k
k
k
k !
... e /2
2
.40528473456935108578... 4
2
... log3
2
1
3
1
311
Likk
k
J102, J117
= ( ) ( )
( )
1
2
1
2 2 2
1
1 0
k
kk
k
kkk k
= 21
52
1
5 2 12 10
arctanh
kk k( )
K148
= dx
x x
dx
e
dx
ex x( )( )
log
1 2 1 20
3
01 1
.405577867597361189695...
2 15 3 520
dx
x
... H
kk
kk
( )
( )
2
1 2 1
... ( )
!
/k
k
e
kk
k
kk
2
2
1 2
210
... ( )( )
arctan
12
2 1
1 11
1 1
k
k k
k
k k
k
.40587121264167682182... 4 3
31240
log x
x xdx
... ( ) ( )2 3
.40638974856794906619... 2 27 3
2 1
2 2
20
1
G
x x
xdx
( ) arccos
... sum of row sums in 1
k kb a
.40671510196175469192... sinh
sinh2
4
1
22
0
1
x dx
1 .40671510196175469192... sinh
cosh2
4
1
22
0
1
x dx
... 5
1
2
1
2 5
3 2
10 1 5
2
2 5 1 5
5 5
20 1 5
5 5
5 5
arccsch arccsch
( ) ( ) (log
1
5 1 55 2 5 1
( )log log( )
= ( )
1
5 20
k
k k
2 .40744655479032851471... ( )
!/2
11
11
2k
ke k
kk
... 1
6 182
1
2 2
40
1
log
log
( )
x
xdx
... 6
6
.40825396283062856150... 11
21
kk k
.408333333333333333333 = 49
120 6
1
6
1
96
21
24
H
k k kk k
.40837751064739508288... 1
164
2 2 2
2 2
1
2
7 4 3 4 3 4
3 4 3 4 41
/ / /
/ /
sin( ) sinh( )
cosh( ) cos( )
kk
... ( )
1
1
1
41
k
k
k
k
... 14 5 e
.40864369354821208259... sinhcosh
1
2 81 1
12 4
1
erfi erfx
dx
x
.408754287348896269033...
1
2
2 )!1(1
6
1
k
k
kk
B
eLi
[Ramanujan] Berndt Ch. 9
=
12
1
kk ke
... e
e dxx2
1
0
2
... 4 3 1
290 4
( ) ( )
=
442
)()3)(2)(1(
kk
kkkk
... 4 3961
2
( )
...
0 12
)3(93/1xe
dx
... e
x xe(sin cos )log sin log
1 1
23
1
dx
... 3
22
... ( ) ( )
1 2 11 2
1
k
k
k
... 2
317
a k
kk
( ) Titchmarsh 1.2.13
.410040836946731018563...
2 1)(
1)(
k k
k
.41019663926527455488...
1
)4(
3kk
k
k
H
... 21
F
k
k
... ( )/
1 11
21
kk
k
e
k
... 11
2
1
2 1
2
20
1
arctan
tan cos
cos
x
xdx
.4106780663853243866...
1
1
2!!
)1(
kk
k
k
...
100 2!
2
)!12(
2!
!)!12(
1
2
1
2 k
k
k
k
k
k
kk
k
k
kerf
e
2 .410686134642447997691...
0 0 )!2(
!)!2(
)!2(
2!
2
1
21
k k
k
k
k
k
kerf
e
... sine
... 3 93
4
1
3 20
log( )
( )
k
kk k
... ( )( ) ( )
11 1
2
k
k
k k
k
... 1
1 k kk !!
... 2
2124
2
4
1
4
( )
kk
= x dx
ex
3
0
2
1
= log logx
x xdx
x x
xdx3
12
0
1
1
= log log1 112
02
0
e dxx
dx
xx
= GR 4.384.12 log(sin ) tan/
x x d0
2
x
... 1
21 1 1 1 1
0
1
cos cosh sin sinh cos sinh x x dx
...
0
2 3/1
)1(
3csch
2
3
2
3
k
k
k
1 .41135828844117328698... SinhIntegral
k k kk
( )
!( )( )
2 1
2 1120
2 .411474127809772838513... 2 318
1 8 8 8 8cos ...
[Ramanujan] Berndt Ch. 22
1 .41164138370300128346...
2 1
2
sinh (log ) sin ( log )i i
... 7G
= 7
17
... 11
131
kk
... 3 3
8
24
1
log log( )
x
xdx
... csclog2
0
11
x dx
x
.4126290491156953842... 1
2
1 1
12
( )
( )
k
kk
5 .4127967952491337838... kk
k
2
1 2 1
... ( )
(( )!)
2
2
1
2
2 212
10 0
k
kI
kJ
kk
... e5
.41318393563314491738...
( )4 1
4 11
kk
k
.41321800123301788416... 2
3
2 3
9
1
2 arcsinh
= 2 11
111
2
1
2
1 22
Fk
k
k k
k
, , ,( )
... 1
331 kk
.413292116101594336627... cos(log ) Re i
2 .4134342304287...
2 1
21
1n k
nk
... 2
3 3 5 52 3 30
/
dx
x
... 1
2 4
2
0
cos cosh( )!
k
k k
6 .41376731088081891269... 16
243 1
4 3
30
log x dx
x
... e
e x x dx
1
102
0
sin cos x
... log12
6 42 1
1
H kk
k
.4142135623730950488... 8
5cot
8tan
2)!2(
!)!12(12
1
kkk
k
... 2
= 1
8
2
0k
k
k
k
= 11
2 11
( )k
k k GR 0.261
.4142135623730950488... 8
csc8
3sin21
.414301138050208113547... 1 2
3
3
6 3
1 3
4
3 3
4
log
( )
i
i
i i
2
6 3
3 3
4
1 3
4
i
i
i i
( )
= ( )
1
1
1
31
k
k k
... 1
31 k
kk
.414398322117159997798...
13)2/1(
1
2
3,38)3(7
k k
8 .4143983221171599978... 7 31
12
31
( )( )
kk
... 11 1
21 1 1 1 1
eE i i i i( , ) ( ) ( ) ( , )
= ( )
!( )
1
1
1
21
k
k k k
... 7
6 2 6
3
2
1
2 322
cot
kk
.4146234176385755681... 4
3
2
9 31111
1
22
1
4
HypPFQ { , , , }, , ,
= k
k
kk
k 21
1
( )
.414682509851111660248... 1
2 pp prime
.41509273131087834417... 1)1(2
22log1
kk
k
k
... 2
e
.41517259238542094625...
1
)3(
3kk
k
k
H
... ok kj
k k jk( ) ( ) ( ) )
1 1 12 11
1
... 11
6 2 22 2 4 4
1
coth cot ( )k
k
k 1
= 4
442 kk
... 25
9 1
2
6 50
log
/
x dx
x
... k kk k k
kk
33 6 3
3 41
3 14 1
1 ( )
( )
( )
... 10
...
( )( )
( )2 1 1
3 36
122 4
22
22
11
f k
k jk kj k
... G E x 1
22
0
1
( )dx Adamchik (17)
.416146836547142386998...
0
1
)!2(
4)1(2cos
k
kk
k
1 .416146836547142386998... 2 1 1 2 14
22
1
sin cos ( )( )
1
!
kk
k k GR 1.412.1
7 .4161984870956629487... 55
...
1
4
1
4
1 1
42,
... arctan 3
3 220
dx
x x 10
.41652027545234683566... 3
16 2 2 12 30
dx
x( )
...
( )34
.416666666666666666666 = 5
12
1
222
k kk
...
( )2 1
4 11
kk
k
.41715698381931361266... | ( )| k
kk 4 11
... 7 3
2 2 84
32
27
1 1
4
2 3
12
( ) ( )( ( ) )
Gk k k
kk
.417418352315624897763... 3
81
4 20
erfi
e k
k kk
!( )3
.417479344942600488698...
1 )2(
)4(Im
kki
k
.41752278908800767414...
k
k
k
k
kk 2
)2()1(
)12(
112log4)2( 1
12
.417771379105166750403...
0
4
222 cossin
23
1dx
x
xxx GR 3.852.5
.41782442413236052667...
1
330 26
)1(
6
2log
36 xx
dx
kk
k
... root of ( )x 3
... 11
1
2
1
1
2 1 1 21
e
e
e ekk
k/
=
1 )1(
11
k
k
kke
e GR 1.513.5
.418155449141321676689... )(Im i
.41835932862021769327... 1
3293 5 7 2
2 2
0
1
( )arcsin arccos
x x
xdx
2 .418399152312290467459... 4
3 3
3
2 2 11
k
k
k
kk
!
( )!! J278
= area of unit equilateral triangle CFG G13
=
1
21
21
)!12(
)!()!(
k k
kk
= dx
x x
dx
x20
3 201 1
/
= dx
e ex x
1
.41868043356886331255... 35
2592
5 63
31
sin( / )k
kk
GR 1.443.5
1 .419290313672675237953...
3
3
2 2 3
1 3
2
1 3
2
2 2
tanh
i i i
i i i
2 3
1 3
2
1 2
21 1 ( ) ( )
= H
k kk
k2
1 1
97 .41935701625712157163... 4 4
.41942244179510759771... 11
2 11
kk
... e
e e
k
ekk
sin
cos
sin1
1 2 121
... sec(log )
2i i
... 1
4
1
2
2
4
1
2
1
2 1 2
1
1
cos sin( )
( )!
k
kk
k
k
148 .41989704957568888821... e e5 5
... 92
3
1
1 3
1
1 3
2 1
2
1
21
( )
( / )
( )
( / )
k k
k k k
1 ...
1
3
)(2 2
)6(
)3(
k
k
k
HW Thm. 301
=
primep p
p3
3
1
1
... 2
0
2
2
loglog sin x x dx GR 4.322.1
...
0 )3)(1(2
12log3
2
5
kk kk
... ( )!
1 11
1
kk
k
k
k
... 1
532 kk
...
8 21
2
30
log
e x
edx
x
x GR 3.455.1
.42073549240394825333... sin ( )
( )!
1
2
1
2
1
1
k
k
k
k
...
02
2/
0
01
cossin)sin(tan)1( dx
x
xdxxxK
= 8
19
...
1 1
0
1 144
14
4
)(
14
12
k kkk
k
kk
k
k
6 .42147960099874507241... e
e
e
k
k
1
.421643058395846980882... ( )( )
12 1
1 0
1
kk
k
k
.421704954651047288875... 2 21
2 220
Jk k
k
kk
( )( )
!( )!
... 3 2
2
9 3
16
6
30
log log sin
x
x GR 3.827.13
.42176154823190614057... e2 2 2 2 2 2 2 2 2 2cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )
=
ie e
k
ke e
k
k
i i
2
2 22 2
1
2 2 sin
!
... k H
kk
kk
2
1 2 2 1( )
... sin51
... 1
21 1n
kn k
n
( )
... H
kk
k
2
31 1( )
1 .4220286795392755555... log log
( )3 2 2
3
2
3 6
2 2
34
1
23 3
Li
=x x
x xdx
log
( )( )
2
21 2 1
...
( )
( ) ( )
3
2 3
...
2
1122
1
2
log( )k
kkk
...
33 1
3 1
1
1 60
log( )
( /
k
k k )
... 1212log4
1
16 22
2
Berndt Ch. 9
... 11
3
1
2
2
2
1
1
( )k
kk
k
...
2
1)()2(1
k k
k K Ex. 124g
=
221
1
1log
11
kk kk
k
kkLi
=
0
logdx
e
xxx
= x
dx
xx
x
0 1
1sin GR 3.781.1
= cos xe x
dx
x
1
10
1 .42281633036073335575...
122
1k
k
k
.422877820191443979792...
5
23Li
.422980828774864995699...
1 )2()!2(
4)1(2log)2(
k
kk
kklCosIntegra AS 5.2.16
.423035525761313159742...
1
21)2(k
k
.423057840790268613217...
012 )23(2
)1(
2
1,
3
5,
3
2,1
2
1
kk
k
kF
.4232652661789581641... ( )
( )3
360
4
31
k
kk
... e
... 2
)3()2(
...
2
12
12
1
2
1
4
12log
iiii
=
13
1)1(
k
k
kk
.423565396298333635... B
kk
k ( ) !
20
.423580847909971087752...
1 2
1)1(
kk
k kH
2 .423641733185364535425...
0
32
)!3(
2
3
3cos2
3 k
k
ke
e J803
.423688222292458284009...
0 )4(2
1
3
322log16
kk k
.423691008343306587163... 1
0
2sinlog22log2
1dxxxlCosIntegra GR 4.381.1
.423793303250788679449...
3
2
1
7
)1(
21
57csc
72 k
k
k
.424114777686246519671... 1
0
32 log)1log()3(6)4(624 dxxx
.424235324155305999319...
2
1)(k
kk
1 .424248316072850947362...
12
12
1 11)1()1(
kk
k
kLi
kk
k
.424413181578387562050...
2/1
1
.4244363835020222959...
2
1
2
121 4
1
1 0
2
eerfi
kk
k
e xk
k
x/
( )
!
sin
dx
.424563592286456286428...
02 15
1
15
7
2
21tan
21 k kk
.424632181169048350655...
2
1)( 1)(k
kk
14 .424682837915131424797... 12 31
2
20
( ) /
x dx
ex
.42468496455270619583...
22 1
2
20
loglog sinx x
xdx
1 .424741778429980889761...
12
1
1 1arctan
12
)24()1(
kk
k
kk
k
=
2
11log
2
11log
2
11log
2
11log
2
iiiii
...
328 3 123 2csc cosh cosh sinh coshh
= ( ) ( )( )
( )
1 2 11
11 2
1
2 2
2 32
k
k k
k kk k
k
9 .424777960769379715388... 3
.425140595885442408977...
2
31
2
3122
3
2 ii
=
1
1 )3()13()1(k
k kk
.425168331587636328439... dxx
x
e
22 1
2cos
.4252949531584225265... ( ) log arctan
12
1
3
3
22
1
21 2
1
k kk
k
H
... 2
coth
...
13
2
42log22log2
k
k
kk
H
.4254590641196607726... csch1
2 1
12 2
0
1
e
e e kk
GR1.232.3, J942
.4255110379935434188... B
k
k
kk
k !
2
0
2 .42562246107371717509... 2
3
7
42 3 2
3 3
1
2 2
32
1
( )( )
k k
k kk
= ( ) ( )( ( ) )
1 12
k 1k
k k k
.425661389276834960423...
1
)2(
3kk
k
k
H
.425803019186545577065...
0 )33)(13(
1
4
3log
312 k kk
.42606171969698205985... ( )( )
12 1
2
1
213
1
kk
k k
k
k k
=
1
21
21
2
i i
...
0 2)!2(
)1(2
4
kk
k
ke
8 .4261497731763586306... 71 ... 1)()1( xofroot
... 1
21 Fkk
...
2
53
2
15log
15 22
2
Li
Berndt Ch. 9
...
0
1
27
)1(
k k
...
1 )32(
1
3
2log2
9
8
k kk J265
1 .426819442923313294753...
G xx
dx
x
3 3
81 1
12
0
( )log( ) log
... 2
22 28
23
2
4 1
2 2 1
1
2
log( )
( ( ) )k
k k
k k
kk
k
... 12log22
...
1
21
)!2(
2)1(
4
2sin22cos2
k
kk
k
k
... dxx
x
4/
0
4
sin1
cos
812
12
... 3
2 21
3
1
21
log( )k
k k k
= dxxx
x
02
2
1
1log
= log( )
11
1 30
xdx
...
1
23
23
2
2 3cos)1(
2
1
124
9
k
kii
k
keLieLi
GR 1.443.4
...
0 22
)1(
2
1
2
22
k
k
k
... x x dtan0
1
x
2 .428189792098870328736...
133/23/1
11
)1()1(
1
2
3cosh
1
k k
Berndt 2.12
=
12
11
k kk
.42821977341382775376... eie
Re2
cos
.42828486873452133902... 1
log!
1
2
k
k
kk
.428504222062849638527...
1 )1(
)1)1)(log(1(
kk
k
e
H
e
ee
.428571428571428571 = 7
3
.4287201581256108127...
11
1
1
1
1
1
1 2eEi
k k k k
k
k
k
k
( )( )
( )!
( )
!( )
=( )
( )! !
1
1 22
k
k k k
... 6e
.42881411674498516061... 2 2
20
1
12
1
2
2
2
2
2
1
1
log log log( )
( )
x
x x
...
2
3
21
1
1 2
0
x
e edx
x x
/
.42909396011806176818...
0
3
)!3(
3
54
172
k
k
k
e
... 23
2
... ( ) ( )1 2 1 2 i i
... 22
=1
4
12 1
43
201 k k
k
kk
( )
=( ) ( )!
( )!
1 1 12
121
k
k
k
k
= log ( ) ( )2 11 2
21
1
k
k
k
kk GR 8.373
=2
22 1
2
2 1 2 2 11 1
k
kk
kk
kk k
k
k k k
( )
( )!!
( )!! ( )
=
sin 4
1
k
kk
= arccos arcsinx x0
1
dx
=
02 cosh)1( xx
dx
GR 3.522
.42926856072611099995...
0 1
log
3
13
2
2log3 xx ee
x GR 4.332.2
...
163
22
22
2 22 2csc cot cot csc
2
= kk k k
kkk
k
2
1
2 2
2 31
2
2
2 2 1
2 1
( ) ( )
( )
4 .429468097185688596497... 7
3
.429514620607979544321...
052 )1(256
35
x
dx
.429622300326056192750...
12
1
2
)1(
kk
kk
k
H
...
23
23
221)(2
kk
k
kkLi
k
k
... G
...
22
1 3
31)2(33cot
2
32
kk
k
kk
... dxx
x
04
22
)1(
log
9
3
...
1 1222 4
1
4
)2(
k kk k
Lik
k
.43026258785476468625... 1
2 1
1 3
231
1
1k
k
k
k
kk
( ) ( )
... 2
5
1
2
2
5
1 5
2
12
1
1
arcsinh
log( )k
k k
kk
=
0
122!)!12(
!)!2()1(
kk
k
k
k J142
=
1 2kk
k
k
F
=
0123 3xx ee
dx
xxx
dx
.43055555555555555555 =
1 )4)(3(
2
72
31
k kkk
k
.430676558073393050670... 2log5
11 .430937592879539094163... 48 64 211 2
2 20
log ( )/x Li x Lix
dx
... 5/16
...
1
0
1
0
1
022212
2log3
4
33 dzdydx
zyx
zyxG
2 .43170840741610651465... 24 4
22
( )
.431739980620354737662... 1
2 6 21 2
2
2
coth ( ) ( ) ( )k
k
k k
= k
k k kk
3
2 22
1
1 1
( )( )
5 .43175383721778681406...
1k k
k
F
H
.43182380450040305811... 1
5
1
2
1
2
1
2 2 1
1
2 11
arctan( )
( )
k
kk
k
k
.4323323583816936541... 1
2
1
2
1 2
103
12
0
1
e k
dx
x
dx
e
k k
k
e
x
( )
( )!
.43233438520367637377... 1
2
1 4
23 11
1
1k k
k 1
k
k
kk
( ) ( )
... ( ) log ( )
1 21
1
k
k
k
1 .432611111111111111111 = 4)2(
144
205H
... 2e
...
0
2
sin
log1
1x
x
...
1
0
1arctan
44arctan
xx ee
dx
ee
...
0
4
4
3
x
dxee xx
GR 3.469.2
...
2
3 1)()3(12)4(6)2(71k
kk
=
24
2
)1(
)14(
k k
kkk
... 81 3
2
189
4 1
2
1 30
1 ( ) log/
x
xdx
= 13
30
1 3
31
arctan /x
xdx
... k
kk
k
2
2 11
( )
... 413
... 1coshlog2
1log
2
e
e
=
ii
ii
21
21
11log
= 2 2 1
2
2 1 22
1
k kk
k
B
k k
( )
( )!
= 1
0
tanh dxx
...
1
2)2(
21
2sin2
k k
=
1 )2(
11
k kk
9 .4339811320566038113... 89
...
1 )7(
11
k kk
... 1)2(1)1(1
kkk
... log3
...
1
34/1
)!2(
!
2
1
128
79
64
45
k k
kkerf
e
... 10log
1log10 e
...
1
2
1
13
)1(
2
1
3csc
32 k
k
k
...
1 2
1
kk k
... ( )2 1
2 11
kk
k
...
01
2
0
2
)!2(
12)2(
6
5)2(
3
2
k kk
kI
eI
e
... 2 21 2 1 2 2 1
0
2 1sinh / / ( )!
k k
k
.4353670915862575299... 1
21 ( )!! !!k kk
... sin cos ( )
( )
2
4
2
2
1 4
2 11
!
k k
k
k
k
...
1
5
2!
)1(
32
23
kk
k
k
k
e
...
1
23
12 )1(2log2
121
k
k
kk
...
1
0 1
cosh
2
12log)1log( dx
e
x
ee
x
...
12
1)1()1(
)3/1(
)1(
6
1
3
2
4
19
k
k
k
.43646265045727820454... 2
22 2
125
2
5
1
5 2
1
5 3csc
( ) ( )
k kk
...
1
2
)!2(52
k k
ke
...
01
2
!
1
!2
kk k
k
k
ke
...
1
2
)2(1
2)1(
kk
kk
k
H
...
2 12
12
12
21
11
log ( )( )
i i
kk
kk
k
= 2 11
2 21
log
kk
1 .43674636688368094636...
3
1
2
3log3log2log
6)2( 2
22
2 LiLi
= log
( )log
2
20
1
02 11 2
x
xdx e dxx
...
22 2
1
4
3
2
7tanh
7 k kk
... 42 18 3
241
12
0
1
2
( )log logx
xx dx
...
2
)3()(3)2()4(k
kkk
=
24
23
)1(
12
k kk
kkk
.437528726844938330438...
4
2 14
)1(
1820
257
142
14csc
k
k
k
...
1
2
!
1)1(
k k
k
...
1
1
!
)1(1)(
k
kkk
e
k
HeeEie
... x
xdx
3
30 cosh
...
logx
edxx 10
1
...
0 )22)(12(
)1(
2
2log
4 k
k
kk
=
1 12
)1(
2
)1(1
k
kk
kk J72
= 1
0
4/
02
arctancos
dxxx
dxx
.43892130408682951019...
coth
2
1
2
1
121
kk
...
1
21)1(
k k
k
...
1 )2(
)3(Im
kki
k
...
2
2
1
2
1)(2
12)1(
)12)(1(
2
3
72log4
kk
kk
k
kkk
5 .43937804598647351638... 7
120
2
4
2
83
1
2 2
4 2 2 4
4
3
21
log log log
Lix dx
x x
...
1
1
2
2
)1(
kk
k
.43944231130091681369... e
e x
dx
x2
5
2
14
1
cosh
... 4 7 6 2 5 3 41
61
( ) ( ) ( ) ( ) ( ) ( )
( )
k
kk
... /1
.440018745070821693886...
0 )13)(2(
)1(
5
2log2
5
1
35 k
k
kk
.44005058574493351596...
0
1 4)!1(!
)1(
2
1)1(
kk
k
kkJ
.44014142091505317044... 2
12sin
1
1
2
1cos)1(
2
1sin)21cos22log(
2
1
1
k
kk
...
1
)22(Im
2tanh
2 kki
k
=
0 sinh
sindx
x
x GR 3.981.1
=
dxee
xxx
sin
2 .44068364104948817218...
02)1(!
1)(1
k
k
kk
eeEi
e
...
12
1
0 2)!(
)1()2(1
kk
k
kJ
.440993278190278937096...
2
3
24
3
Berndt 2.5.15
3 .441070518095074928477...
1
3 1)3(9)2(12)3(6)4(6k
kk
=
242
2
)1(
14
k kk
kk
3 .4412853869452228944...
3
4
23
4
1
4
3
8
5 4 3 4/ /
sin
... 31
13
230
( )( )k kk
= dxx
x
02
2 )3log(
3 .441523869125335258... e Ik
k
kkk
00
11
2
2( )
!
... )4(2)3(3
... )572161log(16
52log2arccsch
4
5
10
55
=
0 5/1
)1(
k
k
k
.44195697563153630635... ( ) ( ( ) ) log
1 1 11
11
1
1 2
kk
k k
H kk k
... 3/13
...
... ( )
! ( )
1
1
1
1
k
k k k k
.44269504088896340736... 1
21
1
2 1 2
2 2
3 1 2 21 21
1 3
1 3 2 31log /
/
/ /
kk
kk
k
k
k k
[Ramanujan] Berndt Ch. 31
1 .44269504088896340736... 1
2log
...
1 2
3)2()()1(
)1()3()2(
k k
k kkk
k
=
12)1(k
k
kk
H
.44288293815836624702... dxx
x
1
0 1
arcsin42
4 .44288293815836624702... 21
4
3
4
=
dx
e
ex
x
1
4/
= log log1 14
0
2
0
x dx x dx2
= dxx
x
02
2 )2log(
=
x
dx2sin1
...
1
21221
)!2(
)12(2)1(1tanlog
k
kkkk
kk
B AS 4.3.73
...
00
2 42
4dxxedxex xx
LY 6.30
= dxxx
xxe x
2/
04
2tan
cotcos
cos12
GR 3.964.2
.443147180559945309417...
2
2 1
)1(
4
12log
k
k
k
k
.443189592299379917447...
12 24
12cot
224
3
k kk
= 1)()1(2
2
kL
kk
k
1 .443207778482785721465...
1
2
)12(
)1(2log2)3(2
k
k
kk
Hk
.443259803921568627450 = )15(8160
3617
.443409441985036954329...
0
2/)12(2/12/1
)1(2/1cosh2
11
k
kk eee
J943
3 .443556031041619830543...
12!
2
4,2,2,2,2,2
3,1,1,12
k kk
k
k
HypPFQ
1 .443635475178810342493... 2arcsinh
.4438420791177483629...
log ( )( )
!2
1
2
1
2
1
1
Eik k
k
kk
... 25
.444444444444444444444 =
1 49
4
kk
k
... , maximum value of ee /1 xx /1
=
0 !
1
kkek
5 .4448744564853177341...
32 3
022 3
( )log xdx
ex
... ( ) ( )( )
( )
1 12
8
2 12
2
31
kk
k k
k kk k
k
1 .44494079843363423391... 2 03
1
3( )( )
k
kk
Titchmarsh 1.2.1
...
( )
( )
2
2 11
k
kk
... 9
3
... 32 3
3 3
2
1
3 21
log
k kk
=
4
3
1
31
1
31
1
hgkk
kk
( )( )
= 1
0
3)1log( dxx
.445260043082768754407...
1
2
65
6log
125
42
125
48
kk
kHk
...
1 3
)1(
2
3log1
4
3
kk
kHk
... )()(2
1)4()2(2
3
2
3
1 24
24
ii eLieLi
=
1
4
2cos
k k
k GR 1.443.6
...
0 )2()!12(
)1()21sin21(cos2
k
k
kk
.446577031296941135508... ( )
( !)
k
kI
k kk k2
20
1
21
1
...
2 )(
)1(1
k k
k
2 .4468859331875220877... k
dkk
1
2
... 2sin22cos2
=
1
1
33 )1()!2(
2)1()()(
k
kkii
kkeLieLi
.447213595499957939282... 5
5
... 128
27
32
3
4
3
24
27
144 2
27
2
G log
=log( ) log
/
11 4
0
1
x x
xdx
.44745157639460216262... 3 3
2 21 2
1 33 3
0
/
( )(
dx
x x )
...
2
23 2
112
3
8
1
21 1
1
2
( ) ( )k
k k
k kk k
2 .4475807362336582311...
2
3
21
32
3
2 3( ) /
.44764479114467781038... 2
3tan
.44784366244327440446... ( )
( )log
1
3 11
1
31 1
k
kk
kkk
...
12 2 61
6
3
2 125 6coth csc ( ) tanh/
i icsch
2
= ( )
1 1
4 21
k
k k k
810 .44813687055627345203... 2sinh2coshsinh2sinh2coshcosh2
1
2
2
eeee
=
0 !
cosh
k
k
k
ke
... 4 2
12 1
2 1 20
cos( )
( )
k
kk
GR 1.444.6
.44841420692364620244... log
log loglog2 2
2 2
2
22 3
3
2
1
3
1
2
Li Li
=
3
263log32log3
6
12
222 Li
=
112
1
2
22
32
)1(
4
1
2
1
2
2log
12 kk
k
kk
k
k
H
kLi
= ( )
( )(
1
2 1 20
k
kk k k )2
=
12
21
02
2
)12(
log
)2(
log
x
xdx
x
x
= log 120
e
dxx
.44851669227070827912...
1
22
2
1
2
)15(
5
5
)2(
5csc
5
2sin52
40 kkk k
kkk
1 .448526461630241240991...
122
12
1 1)2()1(
kk
k
kLi
k
k
...
1 !
1
k kHk
...
1
333
cos
2
1
k
ii
k
keLieLi
...
0
3/1 3)!2(
)1(6
9
kk
k
ke
... 7 1
5 2
70
x dx
x
/
... 32
9 3 2 2
12
12
1
( )!(
( )!
k k
k
)!
k
.449282974471281664465... Li2
2
5
...
1322
1
2
1
x
dx
.449463586938675772614...
18 2
2
9 3
2
9 13 20
log
( )(
dx
x x 2)
... 6
... 41 11
6
1
ek
kk
( )!
...
12 2/14
1
22cot
221
k k
3 .4499650135236733653... 9 6 23
24 32
22
2 3
20
( )logx xd
e x
x
= 20
9
... dxx
xxx
x
x2
0
222
)2(
2log)1(log
3
2log
GR 4.313.7
... log( )
( )(2
3
3 2
1
2 1 3 20
k
k k k )
2 .45038314198839886734...
1
1
!!
2)1(
k
k
k
... 1
2 22 2 2
2 2
2
cosh sin sinh cos
cos cosh
= ( )
1
1
1
41
k
k k
.45066109396293091334... 3 2
2
2 1
2 120
cos sin ( )
( )!( )
k
k k k
...
13
1
2
)1(
8
)3(3
k
k
k
=
022
3
2xx ee
dxx
= 2 3 3 22
0
1
Li i Li i Li xdx
x( ) ( ) ( )
... )2(216
36
.45091498545516623899...
( )2
21
k
kkk
.45100662426809780655...
3
3
0
1
83 6 arcsin x dx
.45137264647546680565...
0
3
3
2
3
1xe
dxx
3 .451392295223202661434... loglog( )
34
1
2
20
x
xdx
.45158270528945486473... log( )
log 2
2
41
1
412
1
k
k kkk k
Wilton
Messenger Math. 52 (1922-1923) 90-93
= iloglogRe
= x
dx
x
x
log1
11
0
GR 4.267.1
= dxxx
2/
0
cot1
GR 3.788
1 .451859777032415201064...
0 !
sinhsin
4
1111
k
eeee
k
kkeeee
i iiii
...
0 212
)1(
13
13log
34
1
12 k
k
k
.4520569031595942854... 4
3)3(
... pk
kk
p prime k
2
1
2
( )
log ( ) Titchmarsh 1.6.1
.45231049760682491399... 3
4 2
3
2
2
3
1
2 3
1
21
csch
( )
/
k
k k
.452404030972285668128... 1
42 4 2
1
coth ( ) ( ) ( ) ( )
i i k kk
4 1
16 .452627765507230224736... 2
22
2 1
2 1
0
erfik k
k
k
!( )
... 1
6
1
2
1
3
1
2
5
6
1
3 20
, ,( )
k
k k
...
1
1)3(
)2(
k k
k
... 22
22log
28
5
2
2log5
28
5
16
51
=
2 8
)(5)1(
kk
kk k
... log2
3 20
1
e e
edx
x
x
... 1
2 ii
... 3 11
2
( )k
k
.453018350450290206718... 2
4
22
0
sin /x
ex
... e
6
9 .4530872048294188123... 16
3
42 2
40
sin ( )x
x GR 3.852.3
...
0
2 3
sin
3log1
1x
x
.453449841058554462649...
4 3 3 4 4 320
21
40
dx
x
dx
x
x dx
x 3
... 4 2
3
1
3 6
3 3
4
3 3
4
logcot
( )cot
(
i i
= ( )
1 1
41
k
k k k
.453592370000000000000 = pounds/kilogram
...
cot log log7
88 2 2 2
2 2
2 2
= 1
4 2 221
3 42k k
k
kk
k
/
( )
.45383715497509933263... sin/
x dx0
4
.45383998041715262... ( ) log ( )
112
n
kn
nk
... cot(log )
i i
i ii i
... 3
4
5
21
1
3 2
0
x
e edxx x
/
.454219904863173579921... Eik kk
k
1
22
1
21
log!
AS 5.1.10
4 .45427204221056257189... k
k kk k
2
21
21
4
21
2
2
.454313133585223101836... ( ) ( )
1 2 11
2
k k
k
... ( )kk
12
...
( )
( )
3 1
4 1
.45454545454545454545 = 5
11 63 2
1
F kk
k
... 2 3 21
21
( ) ( )
( )
k
kk
... sin
( )( )!
2
21
2
2 11
42
02 2
1
kk
k kk k GR 1.431
= 0
2 /
.454822555520437524662... 6 8 22 2
1
1 21
log( ) (
k
k Bkk k )k
... ( )
( )!
/2
11
1
21
2
k
k
e
kk
k
k
... 1
9log
... 22
21
7 31
( )log
( )
x
x xdx
... i
i ii
i4
3
22
3
22
sinh( ) ( )
= e x
edx
x
x
sin
10
.455698463397600878... ( )
logk
k k
k
kkk k
1
4
1 4
4 11 1
.45593812776599623677... i e i / /2 4
.4559418900357336741... 12
sinh
... 9
2
12
q2 q squarefree with an odd number of factors
Berndt 5.30
.456037132019405391328... 1
8
1
4
1
44
1
2i
ii
i itanh tan csc
1
84
1
2
1
4
1
4csc cot cothh
ii
i i
= e x
edx
x
x
sin
1 2
979 .456127973477901989070... sinh
2
2
2
21
1
kk
...
1
2
3
21
3
2 1 21
cos( )
kk
.45642677540284159689...
2 3 3
1
2
1
3 11
2
321
1
1
coth ( )( )
k
k
k
kk
k
... 1
2 2 3 3
1
3 120
coth
kk
= 137
300 5
1
55
21
H
k kk
.45694256247763966111...
( )
( )
2
3 1
1
312
1
kk
k k
k
.457106781186547524401... 1
2
1
4 1
3
0
4
cos
sin
/ x
xdx
... 2 21
2
1
log( )
!
!
k
k kk
... cot 2
.45774694347424732856... log2 1
3
1
12
3 3
2
3 3
2
3
2
3
2
= ( )
1
332
k
k k k
.457970097201163304227... 12
cosh
... G
2
= Adamchik (5), (6) log( sin ) log( cos )/ /
2 20
4
0
4
x dx x dx
= x dx
e ex x2 20
.458481560915179799711...
2 1
4 2112 2
1
2
1
sinh
( )k
k k k
.45860108943638148692... 7
1920
1
4
1
16
4
42
42
44
44
Li e Li e Li e Li ei i i( ) ( ( ) ( )i
= ( )sin
1 14
41
k
k
k
k
... 7 3
42 1
1
21
3
( )( ) ( )
( ) (
kk
k
k k k )
.458675145387081891021... 1 11
11 0
log( )ee k
dx
ekk
x J157
=
1 !k
k
kk
B [Ramanujan] Berndt Ch. 5
2 .458795032289282779606... 6 4 12 3 7 29
81 13
2
( ) ( ) ( ) ( ) ( )
k
k
k k
= 8 5 4
1
3 2
42
k k k
k kk
( )
1
2 .45879503228928277961... 7 2 12 3 6 49
81 13
2
( ) ( ) ( ) ( ) ( ( ) )
k
k
k k
= 8 5 4
1
3 2
42
k k k
k k
1
k
( )
... cos( sin ) (coshcos sinhcos )1 1 1 1
= ee e
k
k
ii e e
k
i i
2 1
2
1
cos
( )!
22 .459157718361045473427... Not known to be transcendental e
... 1
84
2
2 2
2
22
2 0
1
K
K
E xdx
x
( ) GR 6.151
... 7 6
45
3
2
12
51
( )( )
( ) ( )
k
kk
... 1
2
1
2
1
2 1 2
1
1
sin( )
( )!
k
kk k
.4596976941318602826... 21
21 1
1
22
1
1
sin cos( )
( )!
k
k k GR 1.412.1
= sinsin log
x dxx
xdx
e
0
1
1
= sin1
21 x
dx
x
...
1
... kk
kk
( )
( )
1
12
...
312
6 21
3
2
3 21
log( )k
k k k
... 2224
22
=
022 )2)(1( xx
dx
= dxx
xdx
x
x
2/
02
22/
02
2
cos1
cos
sin1
sin
... 9 e
... ( )k
k kk
2
... ( )
( )!!
1
2 1 22 10
k
kk k
...
1
2
... Gx
xdx
log log( )2
4
1
120
= log x
xdx
2
21
1
1
GR 4.295.14
= x
e edxx x
2 20
=
cos sin
cos sin
/ x x
x xdx
0
2
GR 3.796.1
= arccot x
xdx
10
GR 4.531.2
= dxx
1
02x-1
arcsinh
= GR 4.227.10 log tan/
10
2
x dx
1 .46057828082424381571...
2
1
77 22
0
arctandx
x x
... tan
!
k
kk
1
... 1
211
3
22
1
4
1 1
2
1
1
HypPFQk
k
k
k
, , , ,( ) ( )!
( )!
... 1
31 1
3 3
21 1
3 3
25 31 3
1 6 2 32 3
1 6 2 3
//
/ //
/ /
( ) ( )
i i
= 1
31
3 1
32 11
1
1k k
k
k
kk
k
( )( )
1 .46103684685569942365... tanh
!
k
kk
1
.46119836233472404263... )12sin(4)!(
)!2()1(
2
1sin
1cos2
1
02
kk
k
kk
k
Berndt 9.4.19
...
1
1
11
41
4
4
1
k
j
kk j
k
jlim( !)
.461392167549233569697... 3
4 2
1 1
2 120
cos
( )
x
x x
dx
x GR 3.783.1
=
0
5 log2
dxxex x
...
eEi
2
1
2
1 .4615009763662596165... 2
12
( )
( )
k
kk
.46178881838605297087... 9 2 3 12 2
301
14
0
1
3
loglogx
xdx
.461939766255643378064... 2 2
4
... 2 2
3
1
3 1 3 20
log ( )
( )(
k
k k k )
= dxx
x
3
1
0
2 11log
= dxx
x
1
04
6 )1log(
... e
e
B
k
kk
k
1
1
1
22
2 1
2
22
1
tanh( )
( )! AS 4.5.64
...
... 4
27
8 2
9
2
2
G
( ) J309
=
1
22 )23(
1
)13(
1
k kk
=log x dx
x30
1
1
1 .46243121900695980136... 11
2
p
p prime
... ( )
!!
k
kk
2
... ( )/
kk
k
11
2
1
2
1 .4626517459071816088... =
21
1
2 10
erfik kk
!( )
3 .4627466194550636115... 11
2 1
1
1 21 1
kk
kk ( )
... 205
44
1
4 10
log( )
( )(
k
kk k k 2)
... 1
21 2 2 csch
=
1
1 )22()2()1(k
k kkk
= )1(Re)1(
)1( )1(
222
ik
kk
k
= dxee
xxdx
e
xxxxx
00 1
cos
1
cos
... 3
6
...
4/1
4/34/14/3
3
1cot
3
1cot
3
1
4
i
=
1
223
1
k kk
=
k
k k
3
)24(1 1
... 21
22
1 5
2
12
21
21
arcsinh2
log( )k
k k
kk
... 6
.4634219926631045735... e Ei Eidx
e xx( ) ( )( )
2 1
10
1
...
1
8
85764801
319735800
kk
k
.463518463518463518 = 6
13
= 5)3(
3600
5269H
... arctan arcsin1
2
1
5
= Im log( ) Re log2 12
i ii
=( )
( )
1
2 2 12 10
k
kk k
=( )!!
( )!!
2
2 1 51
k
k kkk
= arctan( )
2
2 3 20 kk
[Ramanujan] Berndt Ch. 2
= ( ) arctan( / )
12
11
21
k
k k [Ramanujan] Berndt Ch. 2, Eq. 7.5
.4638805552552772268... e
e
ek
k
k
( )1 1
1
... 3212
...
1
1
1 1
)1(
1
1
kk
k
kk ee
Berndt 6.14.1
...
23 6
)1(
k
k
k
...
2
2 )3()()1()3(4)4()2(9k
k kkk
...
1 )4(!249
k kk
ke
... e9 ... 3/1
.4646018366025516904... 5
4
AMM 101,8 p. 732
.46481394722935684757...
22 1
1
3
4
2
3tanh
3
3
k kk
=
2
35
2
35
3
3 iii
=
1
)3()13(k
kk
... 21
2
3
2
1
2 2 1 2 20
arctan log( )
( )(
k
kk k k )
= log( )
11
2 1 20
xdx
1 .465052383336634877609... 2
2
0
21
1
4 21
sinh
/
i kk
.46520414169536317620...
1
3/1
3!3 kkk
ke
.4653001813290246588...
2
21)(k
k
1 .4653001813290246588...
2
2 )()(k
kk
2 .4653001813290246588... 2
2
1( )kk
... 2 23 3
4 121
12 1
4 31
log( ) ( )
k
k k k
... 5 2log
.4657596075936404365... I
e k
k
k
k
kk
0
0
1 1
2
2( ) ( )
!
... ( ) ( )( ( ) )( )
1 2 1 2 1 1
3 1
11
2
2 22
k
k
k kk
k k
...
( )( ) ( )
11
2
k
k
k k
k
... 1
1
1
1
kk
...
1
1
0k
k
...
3
1
3
222 LiLi
= log log11
11 1
1
k kk
k
.46716002464644797643... 1 2 10
1
arcsinh arcsinh x dx
... ( ) ( )
!/2 2 1 1
11
1
2
2k k
k
k
ke
k
k
k
... 2
1
142 1
1
2
( )( )k
kk
kk
= ( )( ( )k k
kk
1 1
2 12
)
=( )!
( )!( )
k
k kk
1
2
121 2 1
x xdx x d2
0
2
2
0
1
cos arcsin/
x
1 .46740110027233965471... 2
214
12
2 1 1
( )!!
( )!! (
k
k k k )k
2 .46740110027233965471... 2
22
114
1
2
2
2 1
log( )
( )i
k k
kkkk
= 1arcsin2
= ( )( )
12
2
1
kk
k
k
k
!
=
0 sinh x
dxx
= log
( )( )( )
x
x xdx K k dk
1 10 0
1
3 .46740110027233965471... 2
141
2
2 1 1
( )!!
( )!! (
k
k k kk )
... E1
4
... 27 3
162 2
6
50
loglog
sin
x
xdx
... 2 11
1
2/k
k
.4678273201195659261... 11
22
1
kk
...
2 2 1
22
1 420
log
( ) sin( /
x dx
x x ) GR 3.552.9
...
2 2 1
2
1
1 40
log ( )
/
k
k k
...
1 2
)3(
2
722log8
kk
k
...
6
31
18
1
921
coth
kk
J124
... 1
32 3
3
110
Ik k
k
k
!( )!
= 15
32 5
2
1
kk
k
.46894666977688414590...
3
4
3
8
4
20
cossin x
xdx
.46921632103286066895... H kk k
k
kkk k
1
2 2
11
1 1( ( ) )
( )log
=
1
)12()2(
k k
kk
=
1 1
)( 1)1(1n k
kn kH
2 .469506314521047562476... 1
1 kk !
32 .46969701133414574548... 4
3
.469981161956636124706... 2
21
1 .47043116414999108828... 2
51
1
2 21
sinh
( )
kk
= 8
17
... 3
2
9
4
1
2 3
1
1
( )
( /
k
k k k )
...
3
23
5
4
3
322
coth
kk
= ( ) ( )
1 3 2 11
1
k k
k
k
.4709786279302689616... 1
31k
k k ( )
.471246045438685502741...
15
342
)2/1()5(
8
31)3(
2
21
82 k k
k
... ( )( )
( )log arctan
12
2 11
12 2
11
12
1
k
k k
k
k k kk
k
403 .4715872121057150966... 1
4 2
2 22
0
e ee k
ke e
k
sinh
!
...
( )
( )
3 1
5 1
4 .47213595499957939282... 20 2 5
...
Lik
k
kk
3
1
31
1
2
1
2
( )
...
0
2/3
22
3
4
3
2
3
2
5xx ee
x
=
0 13/2xe
dx
.47279971743743015582... 4 3
27
1
3 12 20
dx
x x( )
... 2
3
4
9 3
12 11
k
kk
... ( )( )
( )!sin
12
2 1
1 11
1 1
k
k k
k
k k
k
.47290683729585510379... 6 4 1 4 11
2 1 1
1
1
cos sin( )
( )! (
k
k k k k )
... si
x
dx
x
( )sin
1
2
12
1
... ( )
1
1
1
51
k
k k
... 3
22
1
2 11
2
0
e k
k
k
kkk
kk
! ( ) 2!
GR 1.212
...
4 6
3
2
1
5
1
4 421
tan
k kk 5
... 7
1440
7 4
16
1
2
4 1
41
( ) k
k k
= 9
19
771 .474249826667225190536...
( ) ( )4 1
2744 5
... 7 3
42 2
42
4 2
2 2
21
( )log log
( )
H
kk
k 1
... 2
12
... 2 2 4 3 1 32
( ) ( ) ( ) ( ) ( ) ( )k
k
k k
= k
k kk
3
5 42
1
.47480157230323829219... 26
3 3
1
6
2
1
log
kk k
k
k
...
8
3
3
4
1
4 1
2
20
hg hgx
xdx
sin
( sin )
... 1
1 k kk !
... 1
2 21k
k k
... ( )k
kk 41
... e
ek
kk
k
k
3
5 33 2
5 6 5 61
1
1 3
4 3
31 3
3
2
3
21
3
3
/
/
//
/ /
sin cos ( )( )!
1 .4752572671352997213... x dx
e xx
2
20
... Ai
1
2
... 71 5/
...
12
)2(log)(
k k
kk
...
( )( )
( )
( )
3
2
1
82 1 1
1
12
1
2
2 32
k kk k
kk k
.476222388197391301...
k
k
kk
k 2
)!1(
)1(4,2,
2
1F1
1
1
11
... 3
2 2
( )kk
k
... 3
21
...
1
3 2 33
1
4 121
cot
k kk
... 2
3 2 33
1
322
cot
kk
... erf
1 1
2
... log10 3 5 .47722557505166113457... 30
... 3
2
.47764546494388936239...
sinh
log
=
1
1 )2()1(11log
kk
k
k
kii
... 2
2
2log
[Ramanujan] Berndt Ch. 22
... 8 6 2 7 22 1
22
1
log log( )(
k H
k kk
kk 2)
... 3 3
42
1
4
2
2
2
2
1
2
1
2
( )log
i i i i
= ( )
1 1
5 31
k
k k k
... G
G1
... 4 2
... ( )( )
( )! /
11
2
1
22 1
2
k
kk
k
k
k k
e
... 4
2 11
1
1
2 10
log ( )k
k k k
... 4 3 4 ( ) ( )
.47923495811102680807... 2 2 2
21
2 1
1
21
csc( )
/
k
k k 4
... sin( )
( )!
1
2
1
2 1 22 10
k
kk k
AS 4.3.65
=
4
1
16120
( )
!( )
k
kk k k
... Jk k
kk
k2
1
0
2 2 12
2
( )!( )!
.47962348400112945189... 2 2 11
21
cik k
k
k
( )( )
( )!
... ( )
( )!
/k
k
e
kk
k
k
1
1
2
1
1
...
3 3 2 41 3 30
/
dx
x
.480000000000000000000 = 12
25 4
2
1
Fkk
k
...
2 0 )(
1)(
k k
k
...
2 2
1
10
1
log log log
logx
dx
x
GR 4.229.3
... log( )
( )2
1 1
22 1
2 11
H
k
H
kk
kk
k k
k
= log( )( )
( )
x x
x
dx
x
1 4
2 20
GR 4.299.1
... 2 22 1
2
1
log( )
k H
kk
kk
1 .48049223762892856680... ( )2 1
2 12 11
kk
k
... ( )
! ( )
1
2 1
1
1
k
k k k Titchmarsh 14.32.3
2 .480548302158708391604... 22 e
1 .480645673830658464567... 2 2 1
11
2
2
2k
k
k
k
k
ke
( )
!/
1 .480716542675186062751... 2 3 1 12
213
2
k
k k
kk
k ( )
6 .48074069840786023097... 42
... 2 3
5
( )
... 1
8log
.481008115373192720896... 2 2 2 2 5 21
22
5
4arctan log log arctan log
=
112
1
0 )12(2
)1(
)22)(12(4
)1(
kk
k
kk
k
kkkk
= log( )
11
2 20
xdx
... 0
1 11
1( )
! (
k
k jk jk
)!k
... 2log2 log 24 -8 5 = arccsch
1
22
5
1
cos( / ) k
kk
... 2 2 2 4 31
2
2 21
( ) ( )log
( )
x
x xdx
... ek
k
kk
3 2
0
3
2/
!
... 615 1 3
0
1
ee dx /
x
... ( log )logsin tan
/1 2
22
0
2 x x dx
... 2 3 4 22 1
22
1
( ) log( )
H
k kk
k
... 24 1
2
2
0
loglog( )
cosh /
x
xdx GR 4.373.2
... 15
8
3
2
3
2 3
2
1
logk Hk
kk
7 .4833147735478827712... 56 2 14
... log
( )( )( )
3
2
3
21
k
k km
k m
... ( ) ( )
( )!sin
1 4 2
2 1
1
02
1
k
k k
k
k k
...
1
24
2
6
2 3
3
1
2
1
43
2 3
3Li Li( )log log
= 84 4
2
31
2
0
log x
x x
x dx
ex
...
2
1
41 2 1 2 ( ) (i i
... 4
14
1
2121
( )!
( )!
k
k kk
Dingle p. 70
1 .4844444656045474898... 3 18
2 3
3
3
3
2 2
30
log log log xdx
e x
2
... 3
64
... 2
1
3 1( )kk
... Ei
e
H
kk k
k
( )( )
!
11
1
=
log( )1
0
1 x
edxx
8 .4852813742385702928... 72 6 2
... 5
48 2
2
4
1
2
1
2
2 2
2 2
logLi
iLi
i
= log( )
( )
1
1
2
20
x
x x
...
1 )6(
11
k kk
...
6 3
3
6
1
2 3 13 10
log ( )
( )
k
kk k
Berndt 8.14.1
= dx
x x2 12
... ( )
1
1
1
61
k
k k
... 2 3
3
1
1
3
20
logx
xdx
= log( )
11
3 120
xdx
=
0 cos2
cosdx
x
x
... 4 210
Gx x
xdx
logcos
sin GR 3.791.2
= log( GR 4.224.10sin )10
x dx
... ( ( ) ( )4) 2 3 13
2 11
k
kk
k
9 .486832980505137996... 90
1 .48685779828884403099...
1
)1)()(k
kk
... 10 161
2 30
ek kk
k
! ( )
...
2 23 33 4 40
/
dx
x
... tan/
x dx0
4
... 12 2
1
1 220
csch
( )
/
k
k k
... 2
22 1
( )
( )
k
kk
.48808338775163574003...
15
364 4 1414 143 4
4 4
/ csc csch
= ( )
1
1442
k
k k
.4882207515238431339...
( )k kk
k 21
... 3
42 2
3 3
22
0
1
loglog
log( )x x dx
... 2 2 25
8
1
4 1
4
20
4
arctancos
sin
/
x
xdx
... 2
1 3/
... H k
k
k
kk
k k
1
2
2
2
1 1
2 1
( )log
.48883153138652018302... 2 4 2 2 23
42 log log
( )
= ( ) log ( )1 12
0
1
x x
xdx
... cosh ( )( )
2 21
2
8
22 2 2 2
0
!
e ek
k
k
... 3
5
... 3
22
22
2
1
2
1
log ( )! ( )
e
Eik k k
k
k
... 32 2
2 32 32 2 3
0
( )logx x
edxx
... eG
22 .49008835718767688354...
1
32
2)2()1(13 k
kkk
... ( )
12
1
41
k
k k
kk
... 13
16 12 12 1
2 2
1
log ( )
k
kk
k
Adamchick-Srivastava 2.24
...
2
1
2
40
9
5
2
( )
...
( )32
... 1
2 k kk ! ( )
... 5
32 12 40
dx
x( )
= sin5
30
x
xdx
GR 3.827.9
= dx
x( sin )5 20
2
= x GR 4.523.3x3
0
1
arcsin dx
...
( )
!log
k
k kEi
k kk k
2 1
1 1
k
1
= 5369
36005
6 H ( )
... e i i2 4
... 1
3
12
3
2 3
1
3 20ee
k
k
k
cos( )
( )
!
... dxe
xdx
x
xx
e x
0 022 1
)/sin(
1
cot
1
... 1
6 3 3 1 3
31 3 1 3 2 3 1 3
3
32 ( ) ( )/ / / /
k
kk
...
( ) ( )3 13
21 1
2
1
k
kk
k
...
41
1
2 1
2
0
4
cos
sin
/ x
xdx
... 16e
... 2
123
0
ex
xdx
sin( tan ) GR 3.881.2
...
4 2
2 21
16
1
821
coth
kk
... 2 2 1 2 2 2 1 2 21
2 2 30
arcsinh
log( )( )k
k k
2 .4929299919026930579... 23
2 622 2
0
x xd
ex
log2 x
... 5 3log
... SinIntegralk k
k
kk
1
2
1
2 1 2 2 12 10
( )
( )! ( )
... root of Ei x Ei x( ) ( ) 1
... 3
29 2
2
9
4
9 93
1 3
1 3 1 3 1 3
/
/ /cos cos cos /
[Ramanujan] Berndt Ch. 22
.493480220054467930942...
2
22 1)2(
)2(2
)4()2(
20 nT
,
where n has an odd number of prime factors [Ramanujan] Berndt Ch. 5
.493550125985245944499... 1
2
22
2 k
k
kk
log
.4939394022668291491...
43
3
3 21 0
1
156 2
1
( ) log logx
x xdx
x
xdx
= x
e edxx x
3
0 1
6 .4939394022668291491...
43
156 4 1 ( ) ( ) AS 6.4.2
= log3
0
1
1
x
xdx
GR 4.262.2
= x
edxx
3
0 1
Andrews p. 89
.49452203055156817569...
( ) loglog ( ) log ( )
( )2 2 2
2
3
3
4
1
12
3 2
20
1
x
x x
.494795105503626129386... 2 3 12
2321
k
kk
kk
( )
... 51 4/
... log
log
cos 1 2
0
x
dxx
... G8
... 1
16
1
43
1
2 4
2
20
1
, ,log x
x
=i
Lii
Lii
2 2 23 3
.49566657469328260843... ( )4 2
4 41
2
41
k k
kk 1k k
=
8 2 2 2coth cot
... 1
2 11 ( )kk k
... 3
3 ( )
... 6G
... 2
125
4 53
31
sin /k
kk
GR 1.443.1
.496460978217119096806... 7
720
1
2
1
41 1 1
41 4 1 4 3 4 / / /csc ( ) csc ( )i
= ( )
1 1
8 41
k
k k k
... e
e
dx
x
x
1
4 1
22
20
sin ( )
3 .49651490659152822551... H
kk
k
22
1
...
16
2 2
2
2
21 1
cosIm
( )
( )
k
k
k
ikk
k
...
2 10 2 520
dx
x
... 31 6
32
7 4
8
2
2
1
2 4 2 2
( ) ( ) ( )coth tanh
= ( )
1 1
8 61
k
k k k
...
1
2
1
2
5
4
1
21 4
/
.497203709819647653589... sin
!
k
kk
kk 20
.497700302470745347474... 2 6 2 21
log log log ( )( )
log
k
k kk
Titchmarsh 1.1.9
=
primep p 21
1log)2(log HW Sec. 17.7
6 .497848411497744790930... 7 3
4
3
82
2
16 6 1
2 2 1
2 2
2
2
31
( )log
( )
( )
k k k k
k kkk k
... Im ( ) i
... ( ) ( )
1 14
2
k
k
k
... 12 2 1
22
1
1eEi
H
kk
kk
k
( ) log ( )!
... sin sinh( )
( )
1
2
1
2
1
4 21
k
k k ! GR 1.413.1
.49865491180167551221... k
kk 2 330
... e ek
k
( )
2
... 1
2 22
kk k
( )
... eG
.49932101244307520621...
4
224
2
)1)(log
1dss
kk
k
k
.50000000000000000000 = 1
2
= 1
11 ( )!k k!k
=
1
)12()2(k
kk
= k k k
kkk k
( )
( )
2 1
4
4
4 112 2
1
= 1
3 30
2
1k
kk
k
k
= 1 1
2 122
312k k
k
k k k kk kk
( / )
1
= 1
4 121 kk
J397, K Ex. 109d
= 2 14 3
72
k k
k kk
= k k
kk
2
1
1
2
( )!
GR 0.142
= 1
1 3
1
2)0 1k k k k kk k!( )( ) !(
= 2 1
1 12 21
k
k kk
( )(( ) 1)
K 134
= 1 1
33 11
3 12k k k k k kk k
= ( )!!
( )!! ( )( )
2 1
2
4 1
2 1 2 2
2
1
k
k
k
k kk
J390
= 2
2 30
k
k k k k!( )( )
= ( ) ( ) ( ) ( )
1 1 2 22 1
k
k k
k k 1k
= ( ) ( )2
4
2 1
41 1
k k kk
kk
k
= ( )k
kk 2 11
= ( )2 2
4 11
k k
kk
= ( )sin
( )sin
( )sin
1 1 11
1
12
21
3
311
k
k
k k
kk
k
k
k
k
k
k
=
1
22
241)12(
13 kk
k
kFkk
k
=
122
1)2(exp
11
kk k
k
k
=
122
11
k kk GR 0.261
= 11
2 1
2
2 12
2
21
kk k
k
k GR 0.261
= k k
kk
( )
( )
2
1 21
J1061
= x dx
x
3
40
1
1
= log( )1
31
x
xdx
= log sin log log coslog2 2
11
x x dx x x dxee
= cos sin sin sinx
edx
x
edx
x x
e
x x
ex x x0 0 0
2
0
x
x
= e x x dx
sin cot0
=
0
11
4xe
dxx
= arctanx
xdx3
1
= dx
e ex x
20
= x e dxx3
0
2
1 .50000000000000000000 =
1
2
32
3
kkkF
= ( )sin
131
1
k
k
k
k
=
0
15
4xe
dxx
2 .50000000000000000000 =
1
2
)(2 2
)4(
)2(
2
5
k
k
k
HW Thm. 301, Titchmarsh 1.2.8
=
primep p
p2
2
1
1 [Ramanujan] Berndt Ch. 5
3 .500000000000000000000 =
21
2
32
7
kk
k
... ( )
arctan log1 2
8
1
42
2
16
5 2 2
5 2 2
2
81 2 1 2
3
8
2
16 2 62
arctan arctanlog dx
x x
.500822254752841076485...
( )( )
( ) ( )42
2 12
1
48
1
2 4 4 Li e Li ei i
= cosk
kk4
1
GR 1.443.6
.50112556304669359364... 3
9
1
6
5
3
1
12
7
3 2 42
arctan logdx
x x
.501268927971592850757... ( ) ( )
!!
1
2
k
k
k
k
.50132565492620010048...
0
6
222 cossin
5
2dx
x
xxx
... sin4 1
... 1
1 ( !)!kk
.50138916447355203054... cosh cos cos
( )
1 1
2 4
1
4
1
2
1
4 20 !
e
e kk
2 .501567433354975641473... SinhIntegralx
xdx( )
sinh2
0
2
.50197827418667468842... 1
9 31 k kk
.502014563324708494568... Likk
k7 7
1
1
2
1
2
.502507086021695538022... i i2
16 2
1
2
1
2
1
2
2 2
1 1
( ) ( )
4
2 2
16 2coth
= H
kk
k 4 221
.502676521478269286455... 1
261 k
kk
k
.502733573141073270868... 1 2 3 4 5 6 7 ( ) ( ) ( ) ( ) ( ) ( )
= 1
8 71 k kk
1 .503008799888373907939... H
k kk
k !
1
2 .50307111052515473419... 121
kk
kk
15 .50313834014991008774... 3 2
2 202
x dx
e ex x/ /
.503150623555038937007... 2
1
2 11 2
40
ex x
xdx
/ sinsin
Marsden p. 259
... 1
22 4 6
2
18 6
1
( ) ( ) ( ) cothk kk
...
0 )12(!!
1
k kk
... 1
8 51 k kk
... e e
... 1
2 90 2 2
2 2
2 2
14
8 41
sin sinh
cos cosh k kk
... 1
23
1k
k
... 1
8 31 k kk
... sinh 2
... 11
1211 2 2
5
2
1
4 21
HypPFQx
xdx{ , }, , ,
sin
... Likk
k6 6
1
1
2
1
2
... 11
e k
k
... 8
3
... 2 2 22
0
2
21
Ik
k
k
k
( )( !)
... 675 5 7
12 1
4 5
30
1 ( ) log
( )
x x
x
... ( ) ( ) ( ) ( ) ( )2 3 4 5 6 11
7 61
k kk
.5056902535487344733... sinh3
39001
814
4
kk
... 1
10 2 15
5
3
1
3 520
coth
kk
... ( ) ( )k kk
1 12
...
2 3
)(2)1(3log
331
kk
kk k
.506117668431800472727... 13
125 1 ( )
.506476876667430480956... log arctan3
4
2
2 4 2 22
dx
x x
... 5
2
... 2
... 3
32
1
1 30
log( )
/
k
k k
... 1
3 2 3 32 3 1
1
2
3
23 3 1
1
2
3
22 3 2 3 2 3 2 3 2 3
/ / / / /( )i
ii
i
1
3 21 22 3
1 3/
/
= 1
231 kk
.50746382959658060049... 1
7 41 k kk
... 12
3
1 142 4 4
4 41
csch csch
( ) ( )k i k ik
... logk
kk 22
... ( ) ( ) ( )/ /31
41 1 1 13 4 3 4
i
41 1 1 11 4 1 4 ( ) ( )/ /
= 1
7 31 k kk
.508074963143952841655... 5 2 2 3 ( ) ( )
.50811059680068603330... 1
7 21 k kk
... kH kkk
( )
1 11
...
1
0
1
0
2
1
)1log()3(
4
2logdydx
xy
xy
.508228442902056261461... 1
71 k kk
... ( ( ( )))2
.50835815998421686354... e
e kk3
2
3
3
2 3
1
3 20
cos( )
!
... Likk
k5 5
1
1
2
1
2
...
coth
2 2
4 226
11k
k kk
= ( ) ( ) ( )
1 2 2 21
1
k
k
k k
... (( )!)
( )
k
kk
1
3 1
3
1 !
... 3 22
1
4
1
421 1
log( )
k k
k
kk
k
= dxx
x
1
02
4 )1log(
...
3
2
1
2
1
2
7
4
3
23 4
/
=
3
2
3
4
1
2
1 4
/
... 3 5 4 2 31
41
( ) ( ) ( ) ( )
( )
k
kk
.50914721058334683065...
8 22
2
2 1
2
2 21
csch2
2 sinh( )
k
kk
=( ) ( )
1 2
2
1
1
k
kk
k k
2 .5091784786580567820... cosh cos(log ) sin ( ) 2 2
1
i i
= 11
2 1 20
( )kk
...
12
2
2
2 11
kk k
k
k
kLi
.510572548834680863663... HypPFQk kk
1113
22 2 2
1
4
1
2 21
, , , , , , ,( )!
... 2 4 21
12
340
log( )( )k kk
... 1
1 p kk ( )
...
21 11 2
2
0
2
J J x x( ) ( ) cos(cos )cos/
dx
... log5
3
2
5
2
511
Lik
k
kk
... 1
241 k
kk
k
... 3 2 2
20
2
20
3
8 3 3 1 3
log log logx
xdx
x
xdx
... 1 2 3 4 51
6 51
( ) ( ) ( ) ( )k kk
... 12 2
1
1 2
1
21
csch
( )
/
k
k k
... 1
3
2
3
3
2
1 3
33 3 2
5 6
0
1 3 1 3
e ek
k k
k
/ / /
/
cos( )
( )!
... kk
kk 31
... 1
3
2
3
3
2
3
33 3 2
5 6
0
1 3 1 3
e ek
k
k
/ / //
cos( )!
... 1
31
6 3 3
61
6 3 3
62 31 3
7 6 2 32 3
7 6 2 3
//
/ //
/ /
( ) ( )
i i
2
6
= ( )
( )
3 2
3
1
3 113 2
1
k
k kkk k
... ( )k
kk 2 11
.51301613656182775167... sin ( )
!( )
2 11
20
k
k k k
... ( ) ( )
1
2
1
1
k
kk
k
9 .513507698668731836292... 1
10
... 8
... sinh2
2 21
2
11
kk
... ( ) ( )3 1 3 1 11
k kk
... 285929
11664
1
7 77
7
1
Lik
kk
... 1
7 7loglog e
... ( )!
1 1
1
k
k
k
k
.51394167172304786872...
1 )13(3
32
kkk
k
Berndt Sec. 4.6
=
1
2
3
)(
kk
ku
...
2
4 21
2coth ( ) ( )
= 1
21 2 4 1
11
16 4
1
( ) ( )k
k k
kk k
... 4 4 20
tanhsin
sinh
x
xdx GR 3.981.1
.515121478609336050647... 4 16
2
2
2 2
31
log arctan x
xdx
... 3 3
7
( )
...
( )33
1 3
6
3 3
2
1 3
6
3 3
2
i i i i
= 1
21 3 3 1
11
16 3
1
( ) ( )k
k k
kk k
... log( )
( )
log
( )2
12
1
1 1
2
22
2
30
1
k
k
k
k
x
x
... 3 12
1
k
k
k ( )
... 81 5/
... 1
4 4
1
4
2
4
2
4
i i
i
= cos( / )x
ex
2
10
... F k
kk
3
1 8
... 2
6
2 2
2 2
sin sinh
=
23 4 1 4 1 4
6
1
41 1 1 1 1 ( ) ( ) ( )/ / /
1
41 1 1 1 11 4 3 4 3 4( ) ( ) ( )/ / /
= 1 1
21 4 26 2
1
1
1k kk
k
k
k
( ) ( ) 1
= Im( )
` 12 2
1 k k ik
...
... 1
2
22
2 1
0
2e ek
kee e
k
e
/ coshsinh
!sinhsinh GR 1.471.1
= coshcosh sinhcosh sinhsinh2 2 2 ... ( )3
... 5
64 162
2 2
32
2
20
loglog x
x xdx
...
0
2
xe
dxxx
.51684918394299926361... log
arctan7
6
1
3
5
3
3
6 132
x dx
x
... 1 1
21 5 16
1
1
1k kk
k
k
k
( ) ( ) 1
... 21
4 43
2
21
Lix
x xdx
log
...
2 2
)(
12
1
kk
primepp
k
...
1
1
...
6
1
2 6
1 3
2
1 3
2coth cot
i i
12
2 2 32
2 3i icot
= 1
1
1
21 66
1
1
1kk
k
k
k
( ) ( ) 1
.51716815675825854102... 8
2 3log
... 1
2
21
0
e ek
ke e
k
/ cosh( / )
! J711
= coshcosh sinhcosh coshsinh1
2
1
2
1
2
.5172859549331002461... 1
3 1 333
11 ( )
( )k k
kk k
k
... Likk
k4 4
1
1
2
1
2
... log( )11
e kk
k
... 6 2
2
CFG D1
... 11
31
k kk ( )
... 4 3 2 2 ( ) ( )
... sinh
( )
e
e
e
k
k
k
2 10 !
.51913971359001577609... 2 2
2
1 1 12
1
( ) ( )
!
k
k
k
k
!
...
12
)2(log)(
k k
kk
... 2
19
... ( ) ( )2
2 1
2
2 122
1 1
k kk
kk
k
... ( )2
2 11
kk
k
... H
kk
kk 31
... 1
2
1
1
Eie
Ei ek
k kk
( )cosh
!
... primep
p 22
115
Berndt 5.28
=
1
2
)(
)4(
)2(
k k
k
Titchmarsh 1.2.7
=
squarefreeq
q 2
.519837072045672480216...
1
122 1)2()1(1cschcoth4
1
k
k kk
=
222
2
)1(k k
k
1 .520346901066280805612... 2
10
1e
x
x xdx
e
( log )
/
GR 4.274
.520450870484383451512... 1
4 21 2 1 2 1
1
21 4 1 4
1 4i ii
( ) ( )/ /
/
1
4 21 1 23 4 4 ( ) / /
= k
kk4
1 2
.520459029022087280114... sin sin coshcos sinhcos sin sincos2 2 2 2 e 2
= sin
!
2
1
k
kk
.520499877813046537683... erfk k
k
kk
1
2
2 1
2 2 12 10
( )
! ( )
... log( sin )cos
2 12
1
k
kk
... ( )k
kk
2
2 2
= 25
48 4
1
4
1
4
1
44
21
21
23
H
k k k k kk k k
2 .520833333333843019783... 1
0 ( !)!!kk
...
log log
log(log )2
3
41
42 2 1
2
20
1
x
xdx GR 4.327.1
=
log
cosh
x
xdx
0
GR 4.371.1
.521095305493747361622... sinh( )!
/ /1
2 2
1
2 1 2
1 2 1 2
2 10
e e
k kk
AS 4.5.62
.521113369404676592224... 1
2 2 12
1k k
k ( )
Berndt Ch. 4
... arccot cot csc ( )sin
2 2 2 12
21
1
kk
k
k
k
.521302552157350760576... 2 1 21
21
CoshIntegralk kk
( )( )!
.521405433164720678331... 1
152 2 5 1 2 log( ) , avg. dist. bet. points in the unit square
.5216069591318861249... 1
2
1
2 0 0
e
x dx
e
x dx
ex x
sin cos
... ii
ii
eie
eie
22
22
1sin)1(
= 1)()(sin2
kkk
... ( )2
11
k
ekk
.521934090709712874795... 8
15
1
4 2 1 2
2
2 10
arcsin( !)
( )!
k
k kk
.522507202193083409915... ( )
( )log
k
k kkkk 2 1
1
21
1
212
k
... sinh2
101
42
2
kk
... 1
1442 kk
8 .522688139219475950514... 1
9
... arccot ( cos )cscsin
2 1 121
k
kkk
... 1
21 1 1 1
3
22 2
1
4
12
31
HypPFQk
kkk
, , , , , , ,
.523073127217685014886... 8 340
3
2 1
16 1 20
k
k k kkk ( )( )
... H k
kk
( )3
1 3
.52324814376454783652... 3 3 4 2 11
1 30
log log log( )( )
x xdx
... arctan
...
02 9
2arccsc)1()2(6 x
dx
= x dx
x
x dx
x
x dx
x1 1 1120
2
60
9
120
=
0 3xx ee
dx
... 12 2
1
1
22 0 1
0eI I
k
k
k
k
k
( ) ( )( )
( )!
... 3 3 4 ( ) ( )
... 21
2
1
2 1
3
20
4
arctancos
sin
/
x
xdx
... ( )
( )!!
1
1
2
1
k
k
k
k
... 1
20 ( )!k kk
!
... 2 2cosh
... log7!
... ( ) ( ) ( )4 3 2 11
5 41
k kk
= 1
21 41
1
( ) ( )k
k
k 1
... e
ek
k
k
k
3 2
5 3
5 6 5 63 2
1
1 3
4 3
33
3
2
3
2
3
3
/
//
/
/ //sin cos
( )!
1 .52569812813416969091...
3 2
3 2 1
12
3 1
2 11 2
//coth
( )( ) ( )
k k
k
k 1
... 2
5 5
1
2
3
5 csc
... 1
3
... ( )
( )
k
kk
11 !
...
( )
( ) ( )
3
3 4
= 10
19
... ( )
log
1
2
k
k k k
... H
kk
kk 2 4
1
... ( )
1
150
k
k k
... 5
16
27
5
5
20
logsin
x
xdx
...
1
1
!
)(
k k
k
...
arctan1
1 e
... 7
3
2 1
70
k
k kk
... 1
21 1 1 2 2 1 1
11
( )( cos ) log( cos ) sinsin
( )
k
k kk
...
2
42
7
12
1
21 2 2 12
2
1 1
1
coth ( ) ( )
k
kk
k k
k
=
12 32
1
k kk
.527887014709683857297...
4 2 21
1 2
1
1
log( )
( /
k
)k k k
=( )!
( )!( )
k
k kk
1
2
12
121 2
...
22
1
21log
1)2(22csc2log
kk
k
kk
k
... log8 3
... 1
3
6 3 3
6
6 3 3
61
1
3
7 6 2 3 7 6 2 3
1 3
/ / / /
/
i i
= ( )
( )
3 1
3
1
3 113
1
k
k kkk k
... 24
0
1
ee dxx
... 1
21
12
1
SinhIntegralx
dx
x( ) sinh
... G
... 1
2 1 20k
k
/
...
4 2 2 4
4140
2
3
2 2
34 3 2
2 1
4
log log( ) log
k
k kkk
=9
17
... 1
31 F kk
... k
kk
!
!
12
... 121
kk
k
... ( ) ( ) ( )31
2
1
22 2 i i
= 1 1
21 2 35 3
1
1
1k kk
k
k
k
( ) ( ) 1
... 1 1
22 k
k
kk
... 1
21 ( )!k kk
...
22
2
2
322
2 3 1 1 14 3
1
( ) ( ) ( )( )
k
k k
k kk k
k k
1
... 1
2 11 k kk ! ( )
Titchmarsh 14.32.3
... 8
... ( )( )
log ( )
1 21
1
k
k
k
kk
... ek
k ee
kk
1 1
1
/
!
... 3
2 2 1 3
3 4
40
/
/
dx
x
... 82
36
2 1
12 1 20
k
k k kkk ( )( )
...
2
3
2
20
1
12
1
2
1
3
3
83
1 3
Lix
x xdx( )
log
( ) ( )
... 41
22 1 2
1
1 4
1
1
log( )
/
k
k k
... 34
3
2
2
21
k kk ( )
... 2 1 2 20
4
log cos log(sin )/
x x
dx
... 7 1
3
3
1
3 2
1
e k
k
k
k
( )!
...
3
16
1
61 3
3 3
21 3
3 3
2
2
( ) ( )i
ii
i
= 1
21 3 2 1
11
15 2
1
( ) ( )k
k k
kk k
... ( )k
k
1
2
1
... 62 1623
4
1
ek
k kk
!( )
... log log ( )3 2
3
2
3
2 2
32
1
2
3 3
2
Li
=log
( )( )
2
1 1 2
x
x xdx
1 .53362629276374222515... e
.5338450130794810532... 2 1
2
( )
( )
k
kk
.53387676440805699500...
1 2
12sin
2
1
2
3sin2
2
1sin
51cos4
1
kk
k
1 .53388564147438066683... | ( )| k
kk 2 11
... 1
8
... 9
8
... 16
3
4
31
41
1
log
Hkk
k
.53464318757261872264... log x
xdx
2 121
... 1
2 44 1 4
0
e ek
ke e
kk
/ / cosh
!
.534799996739570370524... log 2 2
... logcos
sin
/
11
2 10
4
x
xdx
...
1
4
1
2
1
2
1
2
1
2
i i i i
= 1
21 4 1 1
11
15
1
( ) ( )k
k k
kk k
... sinsin
coshcos
sinhcos
sinsin(cos ) /1
2
1
2
1
2
1
21 2
e
= sin
!
k
k kk 21
GR 1.449.1
.5351307235543852976... 3
31
1
3 21
sin
kk
... 1
12 k kk !( ! )
... 1
21 F kk
... ( )31
3
... Ai( )1
... 2 2 3( ) CFG D1
... 1
21 5 1
1
11
15
1
( ) ( )k
k k
kk
.5360774649700956698... ( )2 1
2
1
2 12
0
dx
ex
...
e
... 1
21 6 1 1
11
16
1
( ) ( )k
k k
kk
k
.5365207790738756208... 1
3 1 322
11 ( )
( )k k
kk k
k
... 1
21 7 2 1
11
1
2
71
( ) ( )k
k k
kk
k
... 1
31 1
1 3
21 1
1 3
22 3 1 3( ( ) ) ( ( ) )/ /
i i
1
= ( ) ( ) ( )
1 3 1 31
1
k
k
k k
... ( )jkkj
2 111
... k
kk 4 11
... 1
2 2
2
21
sinh
Re ( ) i
... 7 3
8
2
6
2
12
1
2
1
2
3 2
3 31
( ) log log
Likk
k
=log( / ) log1 2
0
1
x x
xdx
...
0
3/1
)!3(2
3cos
3
2
3
1 2/)3/1(3/1
k
k
kee
... 5
2
1 5
2
1
2
1 2 2
2 1 2
1
2 11
log( ) ( )!!
( )!!
k
kk
k
k J141
... H ( )/
31 4
... sin!
1
1 kk
=7
13
... 2 2 2coslog i i2
.539114847327011158805... 1
8
1
4
1
44
1
2i
i itanh tan csc
i
1
84
1
2
1
4
1
4i
i ii
icsch cot coth
= e x
edx
x
x
cos
1 2
.53935260118837935667...
( / )
( / ) ( / )
( )1 2
5 4 3 41
1
21
k
k k J1028
... 1
31
4 2 3 6
21
4 2 3 6
22 31 3
1 3 5 6 1 32 3
1 3 5 6 1 3
//
/ / //
/ / /
( ) ( )
i i
2
32 6
1 3
2 31 3
/
//
= 6 3 16
613
2
k
k k
kk
k ( )
9 .5393920141694564915... 91
... e Not known to be transcendental
... 3 3log 1
... cos cosh Re( )
( )!1
1
20
i ek
ik
k
AS 4.3.66, LY 6.110
.540345545918092079459... 4 4
1
2cosh
.540553369576224517937... 2
21 28
21
2
4 1
2 1
1
2
log( )
( ) ( )k
k k
k k
k
k
kk
... sin(cos sin ) coshcos
sinhcos
1 12
2
2
2
= ( ) sin
!
1 2
2
1
1
k
kk
k
k
6 .540737725975564600708... 37
4 2
2
8
3
0
k
k
kk
k
... ( ( ))3 ... G7
8 .54097291002346216562... 3 2 3 31
3
30
1
( ) ( )log
( )
x
x
... 13 2 10 20 1
6
21
I Ik
kk
( ) ( )( !)
... 4 2
0
1
180
1
log( ) logx x
x
.5411961001461969844... 2 2
2 8 81
1
4 21
cos sin( ) k
k k J1029
.54123573432867053015... dx
x( )0
1
...
1 !
)1()1log(
k
kk
kk
Be [Ramanjuan] Berndt Ch. 5
... B
kk
k ( !)20
... 1
20k
k !
... log ( )( )3
2 6 3
4
81 3
1
3
2
3
26 3
27
31
= k k k k
k kkk k
2
2
2
313
36 9 1
3 3 1
( )
(( )
... 2 1
20
22
83
4
1
3 4
1
4
Gk
k k
kk
k
( )
( / )
( ) ( )
... 2
cosh
... 112 96 21 2 3
21
log( ( )(H k k kk
kk
)
... 1
2
1
2
1
2 1 21
sinh( )!
k k
k
... p
pp prime !
...
2 4 2 8 2 2
2 2
coth
i i i
i i
81
21
21 1 ( ) ( ) i
= H
kk
k 4 121
... 2
11
1 1sinh
)!2(
1
2
211cosh
x
dx
xk
ee
k
...
0
1
)!2(
1
2cos1cosh
k k
eei GR 0.245.5
= 14
2 12 20
( )kk
J1079
... 1
21 1 12log( ) log( )(sin cos ) e e e ii i i 1
= sin k
kk
11
...
1
)2(log)(
k k
kk
8 .5440037453175311679... 73
...
2tanh
2coth22
sinh
2
ii
...
1
)2(log
k k
k
... log
log sin( )/2
44
0
4
x dx
.54445873964813266061... ( ) ( )k kk
2 11
.5456413607650470421... dxe
eerf
ex
x
1
4/1
2
2
11
2
... 7
609 3
1 1
4 4
31
4
30
1
( )log
( )
log
( )
x
xdx
x x
xdx
1 ...
2
1
22 2 2 32
0
arctandx
x x
... ( ) ( )
1 112
kk
k
F k
1 ... F kkk
12
1 ( )
... 1 3 220
1
log log( )
/e
e
e edx
x
x
... 27 3
2 1
2
20
1
Gx dx
x
( ) arccos
... 2
118
1 1
2
( )!( )
( )!
k k
kk
!
= 2
1arcsin2 2
= dxx
x
0
3)1log(
11 .548739357257748378... i isin( log )2
.54886543055424064622... 2 1
1 2
1
2 23
2
1
4
1
2 1F
k
k
k
k
, , ,( ) ( !)
( )!
...
1 )12(2
1
kkk k
... log( )
( )
( )
( )2 1
2
0
01
1
2
1
1
Titchmarsh 2.12.8
... log
( )
3
2
1
2
1
2 2 12 10
arctanh kk k
AS 4.5.64, J941
= dx
x x
dx
x
dx
ex( )( )
1 3 102
2 0
2
=
12)2(
log
x
x
.549428148719873179229... 1
41 1 21 4 3 4csc ( ) csc ( ) cos( cosh(/ / ( 2
( ) sin ( ) ( ) sin ( )/ / / / )1 1 1 13 4 1 4 1 4 3 4
= ( )
1
140
k
k k
7 .5498344352707496972... 57
5060 .549875237639470468574... 8
157! 8 7
1
81
7
0
1
( ) ( )log( ) x
xdx GR 4.266.2
2 .550166236549955... ( )/
1 11
31
kk
k
e
k
... 3
20
... 6
G
... 3 6
... 6
6
1
2 12
( )
... 1
2
1
2
12
22 2
0
coth( )kk
...
1
2
2
)(
kk
k
... 4 2
2
1
1 12 2e e sinh sinh cosh
J132, J713
.5516151617923785687... ek
kk
13
6
2
1 ( !)!
.55188588687438602342...
0
1
13
13log
34
1
10
1
121012
)1(
k
k
k
.5523689696799021586... 8
1 11
4 31
erf erfix
dx
x
cosh
...
1 )5(
11
k kk
... sin
11
21 kk
... 11
5
1 21
21
( ) ( )!
( !)
k
k
k
k
... 2 2 2 11 2
2 10
sin cos( )
( )!( )
k k
k k k
2 .553310333104003190087... 0
2 1
( )
!
k
kk
... arctan 2
2 220
dx
x x 5
... 2 1
21
k
k kkk
!!
1 .55484419730133345731...
1
2)12(
21
2cosh
3
1
k k
...
4 2 820
dx
x
.5553968826533496289... 1
2
1 1
2 0
1
sin cos cos
e
x dx
ex
=coslog x
xdx
e
21
... 2
6
2
3
2
6
2 2
2 21
124
0
1
loglog logx
xdx
.55663264611852264179... 1
2 4 4Li e Li ei i( ) ( )
= ( )cos
1 14
1
k
k
k
k GR 1.443.6
... 21
2
1
2 1 1 21 21
log / /k
kk
...
0
2
)2(
)41(4)1(1tan
k
kkkk
k
B
6 .55743852430200065234... 43
.5579921445238905154... H k
kk
2
1 4
4 2
3
3
log log
... 1
6 6loglog e J153
.5582373008332086382...
132k
kk
k
H
... 1 2 2 22
2 10
cosh sinh( )!( )
k
k k k
... ( ) logk k
kk
1
21
... log
( )! ( )!
log
!
k
k n
k
kk n
n
k
1
1
12 1 2
... Jk
k
k0 221
2( )
( )
( !)
... 2222
3
1
ek
k
k
k
!
...
( )
( )
k
kk
1
12
... 1
21 k kk !!( )
...
1
1
11 4
3)1(
4
3
7
3
4
7log
kk
kk
kLiLi
... 2739 3
2
1
1 3
1
1 3
1
3
1
31
( ) ( )
( / )
( )
( / )
k k
k k k
= 14
251
21
2
1
2
0
( )sink k
kk
x
k F x x
edx
... 1
2 2 11 k kkk ! (
)
Titchmarsh 14.32.3
.56012607792794894497...
1 3
11
kk
=
12/)3(2/)3( 22
3
1
3
1)1(1
kkkkk
k Hall Thm. 4.1.3
... 9 121
3 11 3
0
ek k kk
k
/
! ( ( )
2
... 10
34 2 1
1
2
2
2 122 2
log ( )( )
( )k
kk k
k
k k
... 0
21
( )
( !)
k
kk
...
13
)2(3
)3/1(
127
3
1
2
127)3(13
33
2
k k
...
1
2
1
3
121
33
0
( )
( )kk
... 4 3 3 4 ( ) ( )
...
0 !
)1(
k
kk
ke
= k
k
ek
/1
1
11
KGP ex. 6.69
=n
nn loglog)(lim
HW Thm. 328
23 .5614740840256044961...
4 2 22 4
0
3
208 3
( )log xdx
ex
.56173205239479373631... 1
22 12
2
1
e ek
kk
cos cos(sin )cos
!
... 3 3
0
12
8
9 3
16
log ( ) arcsin
x
xdx
= 9
16
= k kk
k kk k
( )( )
2 1 12 1
12
2
2 22
... 1
22 2 1
1 1
1
log cos( ) cos
k
k
k
k
... ( ) ( )2 4
.56264005857240015056... 1
2 21
1
4 121
sin
kk
J1064
... ( ) ( )
1 21 2
1
k
k
k 1
... 2 23
2 23 2 2
22
1
log log( )kH k
kk
... pd k
kk
( )
21
.5634363430819095293... 6 21
4 300
ek k
k
k kkk !( ) !( )
= 1
1 30 ( )! !k kk
... 1
4
1
6
2
3
11 11
32
0
( ) ( ) ( )
( )
k
k k
... 1
2
1
2cosh
1 .56386096544105634313... 6 2 6 43
3
3
8
6
1
2
42
( ) ( )
( )
( )
k
kk
= ( ) ( ) ( )( ( ) )
1 1 12
k 1k
k k k k
.56418958354775628695... 1
2120
k
k kk ( )!
... 17
16
8 7
0
x
xdx
sinh GR 3.523.10
...
3
1
22
1
4 Berndt 8.6.1
... ( )( )
log
11
1
11
1
2
k
k
k
k k
k
.56469423393426890215... sin k
k
kk 21
... erfik k
k
k
22 2
2 1
2 1
0
!( )
... 11
31
31 1
kk
kk
Q k( )
...
2
2
0
tanhsin
sinh
x
xdx
... 61 4/
... ee
ee
ee
11
11
11cosh cosh int sinh int
1
2
12
1
2
12
1
2
12e
ee
ee
ecoshint sinhint log
= H
kk
k ( )!21
.565159103992485027208... I 1 1( )
... 1
2
1
2
1
2
42 2
1
Lii
Lii k
kk
cos /2
...
1
31
33
cos)1(
2
1
k
kii
k
keLieLi
... 2 24 1
2
20
4
arctancos
sin
/
x
xdx
...
cos cosh( )
( )!2 2
1 64
4
1
0
k k
k k
... 1 2 2 12
41
6
1
cos cosh ( )( )!
kk
k k GR 1.413.2
... e e eEi Eik H
kk
k
1 1 1 11
2
1
( ) ( ) ( )( )!
...
2
33
3
9
3
9
2
3
1
33
2 3
3Li Li( )log log
= 83 3
2
31
2
0
log x
x x
x dx
ex
... 2 2 2 12
11
21
Jk
kk
k
k
( )( !)
... 8
2
4
1
4 1 4 20
log
( )(k kk )
... Ik k
k0 2
0
21
2( )
( !)
... ( )k
kk 31
... 1
6
... 4
.5665644010044528364... e Ei Eix
edxx
21
0
1
2 1 21
( ( ) ( )) loglog( )
9 .5667536626468872669... sinh
2
21
22
1
kk
... k
kk
!
( !!)30
... log ( ) ( )( )15
8
3
21 1
1 3
22
k
k
k
k
k Srivastava
J. Math. Anal. & Applics. 134 (1988) 129-140
... arccot2
... 3 62
3
1
3 10
log( )(k
k k k 2)
... 1
21 ( )k kk
... ( )e
... log( )
log
k
k kkk k2
12 1
1
212 1 k
... 1
2
1
21
2
121
1
e k
kk
k
( )( )!
5 .56776436283002192211... 31
... 2 32
3
4 2
3
2 1
4
2 3
31
( )log log
k
k kkk
...
2
4 216 2
1
2
1
cothk kk
= 1
21 2 2 1
11
12
1
( ) ) Im( )
k
k k
kk k i
5 .568327996831707845285... 3 2/
.568389930078412320021... 1
2 2
1
22 12
( )kk
kk
jkj
... 3
41
3
2
2 3
0
1erf
ee dx
/
x
...
2 1
2 112
00
( )
( )!!( )
k
k kH
... ( )( )
12
1
1
kk
k
k
... e
... 1
31
6 3 3
61
6 3 3
64 31 3
7 6 2 32 3
7 6 2 3
//
/ //
/ /
( ) ( )
i i
1
31
1
34 3 1 3/ /
= ( )3
3
1
3 113
1
k
kkk k
.569451751594934318891... 4 2 5 3 ( ) ( )
... 1)2()(sinh1
kkk
.5699609930945328064...
( )
( ) ( )
log log2
2
1
2 122
2
k
k
p
pk p prime
Berndt Ch. 5
... Eik kk
k
1
22
1
21
log!
... H o
k
kk 31
... coshcos sinhcos cos sinsin sin (coshcos sinhcos cossin )1 1 1 1 1 1 1 1
= sin
( )
k
kk
11 !
2 .57056955072903255045...
1
2
2
)(
kk
k
.57079632679489661923... 2
12
1
sin k
kk
= 2 1
4 2 1
2 1
2 2 11 1
k
k k
k
k kkk
k
( )
( )!!
( )! ( ) J388
= ( )
/
1
2 1
1
21
k
2k k
= arccos
( )
x
xdx
1 20
1
= tanh x
edxx
0
= x dx
x x( ) sinh /1 220
GR 3.522.7
= x x
x
log( )1
1 20
dx
GR 4.292.2
1 .57079632679489661923... 2
i ilog
= ( )
/
1
1 20
k
k k
= k
kk
!
( )2 10 !!
K144
= 2
22
2 1
2 1
4 2 11
2
0 0
k
k
k
k kkk
kk
k
k
k
k k
( !)
( )! ( )
= ( )!!
( )!!
2
2 1 21
k
k k kk
= arctan( )
2
2 1 20 kk
[Ramanujan] Berndt Ch. 2
= arctan( )
1
2 1 5 21 kk
[Ramanujan] Berndt Ch. 2
= 4
2 1 2 1
2
1
k
k kk ( )( )
Wallis’s product
= dx
x
dx
x
dx
e ex x20
2 201 1 2
( ) 2
= dx
x x20 1 2
/
= sin sinx
xdx
x
xdx
0
2
20
= x
xx dx
2
2 20
1
1
( )log GR 4.234.4
= cos GR 3.631.20
/2
0
2
x dx
= arccosh x
x21
= log( GR 4.227.3 tan )/
e x d0
2
x
= 1
2 20 e ex x
= xx
dxlog 11
40
= ci GR 6.264 x x dx( ) log0
2 .570796326794896619231... 2
1221
k
k k
k
3 .570796326794896619231... 2
2220
k
k k
k
.571428571428571428 = 4
7
1 .57163412158236360955... 1
2 32 k kk
/2
.57171288872391284966...
0
1
13
13log
2
1
12212
)1(
k
k
k
.571791629712009706036...
1
2
1
3
1
3
2 1
31
( )kk
k
= 1
3 121 k kk ( )
3 .571815532109087142149...
2
3 1)12(128
111
4
)3(9
k
kk
.572303143311902634417... 4
91
4
3 41
logkHk
kk
.572364942924700087071... log1
2
=
e
e
dx
x
x
x
2
0 2
1
1 GR 3.427.5
...
2
112
1
42 2
k k kk
( ) ( )1
= 1
2 2 2Li e Li ei i( ) ( )
=
223 1k kkk
k
= ( )( ) ( )
11
21
2
k
k
kk k
= ( )cos
1 12
1
k
k
k
k
...
22
2
14
3
12 k
k
k
H
.57289839112388295085... 4 2
8
4 2
20
sin x
xdx
... 3 3
23 2
( )( ) ( )
... 11
21
( !)kk
.574137740053329817241... 2 2
216
1
2
cos k
kk
... 1 2
3
1
31 1
11 3 2 34
1
( ) ( )/ /
k kk
= 1
21 3 11
1
( ) ( )k
k
k 1
5 .57494152476088062397... e ee k
k
1 1
1
/ !
... 10 3
.57525421205443264457... x x
xdx
sin
cos
/
2 20
2
.57527071983288752368... 2
23
30
1
43 2
3 3
8
1
log
( ) log ( )x
xdx
... 16 12 31
2
0
1
( )
log x
xdx
... 12
1
2
1
21 40
e
erfik
k
k
k/
( ) !
( )!
1 .57570459714985838481... log
3
4
...
21 1
2
1
21 1
41 1 ( ) ( ) ( ) ( )( ) ( )i i
ii i
= ( ) ( ) ( )
1 22
k
k
k k k 1
... 3
30
4
321
2 1
2 1
( )cos( )
( )k
k
k
k Davis 3.37
...
21 1 21
1
coth ( ) ( )
k
k
k 1
= 1
14 2 42
2 1kk
k k
( ) ( k ) J124
= sin x
e ex x
10
... Jk k
k
k
k
k
k
k1
0
1
20
21
1
1( )
( )
!( )!
( )
( !)
... 11
41
k kk ( )
... (Euler's constant, not known to be irrational)
= ( ) ( )
log
1 11
1
2 1
k
k k
k
k k
k
= k
kk
k kk k
11 1
1 1
12 2
( ) log Euler (1769)
= 1
11
21 n
k
k k
n
kn !
log
( )
Shamos
= log log1
0
1
xdx Andrews, p. 87
= GR 4.381.3 log sinx x d0
2
x
= 1 2
0
2
sec cos(tan( )tan
/
x xdx
x
GR 3.716.12
= 1
22
1 12 20
x dx
x e x( )( ) Andrews, p. 87
= ( )/1 1
0
1
e edx
xx x Andrews, p. 88
= 1
1
1
0
e xe dxx
x GR 3.427.2
= 1
10
xe
dx
xx GR 3.435.3
= 1
10
xx
dx
xcos GR 3.783.2
= 1
1 20
xx
dx
xcos GR 3.783.2
... GR 6.264.1
10
si x x dx( ) log
= Ei GR 6.234 x x dx( ) log
0
.577350269189625764509... 3
3 6 tan
= 4
3
1
2 1 121 ( ) /kk
9
GR 1.421
= 2 1
6 10
k
k k kkk
( )( 2)
...
( )
( ) ( )
2
2 3
.577863674895460858955... 2 1 2
0e
x
xdx
cos
AS 4.3.146
= dxx
xx
021
sin
= GR 3.716.5 cos(tan )cos/
x x2
0
2
dx
= cos(tan )sin
xx
xdx
0
GR 3.881.4
= x x
xdx
3
2 21
sin
( )
Marsden p. 259
... 27 3
6 6
3
2
1
3
2
3 21
log
k kk
.578169737606879079622... ( )
( )
2
2
1
4
12
12 2
1
k
kLi
kk k
...
(sin sinh )
(cosh cos )
2 2
2 2 2 2
1
2
1
141
kk
= 1
21 4 1
11
12 2
1
( ) ( ) Im( )
k
k k
kk k i
... 3
2
( )
!
k
kk
... 4 log
= 11
19
.57911600484428752980...
2 4 2
12
412 24
7 3
( )( )
k
kk
1 .57913670417429737901... 4
25
2
=1
5 1
1
5 2
1
5 3
1
5 42 2 2 21 ( ) ( ) ( ) ( )k k k kk
6 .57925121201010099506... log !6
3 .579441541679835928252... 3
21 2 2
3
21
9
4 6
2
22
2
log ( ) kk kk k
... 1
12 k kk ( ! )
.5795933814359071842... 1
3 1 31
11 ( )
( )k k
kk k
k
... 2
32
2
1
2
20
2
e e
ex dx
x x
x( ) GR 3.424.5
... 2
3 1
22
20
log( )
xdx
x GR 4.261.5
.57975438341923839722...
3
22Li
4 .579827970886095075273... 5 G ... ( ( ( ( ))))2
... k
kk
k
k
k5
1 11
1
21 5 1
( ) ( )! 1
... e
k
k
kk
1
.58043769548440223673... arctan(tanh ) ( ) arctan4
11
2 11
0
k
k k
[Ramanujan] Berndt Ch. 2
.580551791621945988898... ( )( )
13 1
1
1
kk
k
k
... 2 21
0
e k
k
/ !
... 3
43
1
3 13
2
0
1
( )log
( )( )
Lix
x xdx
3
...
036/13 33
2
x
dx
... k
kk
k
k
k
2
61
1
11
1
21 6 2
( ) ( ) 1
.581824016936632859971... 29
64 4
1 4 3
1
e k
k kk
/
!
... k
kk
k
k
k
3
71
1
11
1
21 7 3
( ) ( ) 1
...
12/1
1 /112
11
1
1
kk
kk k
eee
[Ramanujan] Berndt Ch. 31
=
1 !
)1(
k
kk
k
B
= dxex
x
12)1(
log
... e
e e
B
kkk
kk
k
1
1 1
0 0
( )
!
.5822405264650125059...
122
22
2
1
2
1
2
2log
12 kk k
Li
J362
= ( )
1 1
1
k k
k
H
k J116
= log( )
( )
log1
10
1
21
2
x
x xdx
x
x xdx GR 4.291.12
= logx
x
dx
x
12
=
log( )/ 1
0
1 2 x
xdx GR 4.291.3
=
log 1
20
1 x dx
x GR 4.291.4
=
log
1
2 10
1 x dx
x GR 4.291.5
4 .58257569495584000659... 21
.583121808061637560277... 2G
... sin log1 2
= 7
12
.58349565395417428579... ( )( )
12 1
1 1
1
kk
k
k
10 .583584148395975340846... 3 1
2
2
1
2 2
0
e k
k
k
k
( )!
.5838531634528576130... 2 1 2 12
22 1
2
1
cos ( )( )
!
kk
k k GR 1.412.2
... 101 5/
1 .58496250072115618145... log2 3
.5852181544310789058... 16 8 212
0
1
loglog( ) logx x
xdx
...
2)(2
1
kk
... ( )
1
130
k
k k
... 2
1 2 10
4
dx
xsin
/
... 7
3
4
3
4
31
3 4
0
1
,
/
e dx x
... 0
1 2
( )
( )!
k
kk
...
8
9
... 21
... 1
3
2
3 32 1
22
1
1
arccsch ( )kk
k k
k
...
0 )!2()!1(!
11},3,2{{},
k kkkHypPFQ
... G
... 512
63
5
1 2
/
... 8
tanh2
...
0
32/
)!3(2
3cos
3
2
3
1
k
k
kee
... log( ) log( )
log
k
k
x
xdxk
k
1
2
2
1 0
1
GR 422.3
...
12
2
)1(
)13(3
16
3
)(
363
2
9
1
2
3log
kkk kk
kkk
...
7
8
.58760059682190072844...
132
1
1cosh
)!2(2
1sinh
x
dx
xk
k
k
...
3 2 25 6 60
/
dx
x
... sin5
= 17
10
... 9 3 32 1
6
2
0
k
k
kk
k
... i i i ik i k ik
( ) ( )( ) ( )( ) ( )
1 12 2
1
1 11 1
... ( ) ( )
1 21
1
kk
k
F k 1
... 3
2
3
2 3
1
1 3
1
21
csch
( )
/
k
k k
...
02 2)!2(!
122
kkkk
I
... 3 2
208 1
e
x
exdx
log
... 3
16
23
21
21
k k
k kk
( )
( )( ) J1061
=
032
02
6
0
5
)1(
sinsin
x
dxdx
x
xdx
x
x GR 3.827.12
.5891600374288587219... 8 2 2 4 21
23 21
( ) log( )
/
k
k k k
6 .58921553992655506549...
6
7
... e erfk
k
k
12
12
0
( )
!!
.589255650988789603667... 4/
0
3cos26
5
dxx
... log2
2 2
kk
k
1 .58957531255118599032... log
1
4
... e
xe
xdx
21 2
0
log log
.589892743258550871445...
)62(2
2
6324)31(
6
1 3/13/16/53/1
3/5 i
i
2
6324
6
)31( 3/16/53/1
3/5
ii
=
23 6
1
k k
.59000116354468792989...
12
2
)!2(
),2(
k kk
kkS
... ( ) ( )
arctan
1 1
2 3
1 1
23
2
k
k k
k
k k
k
.59023206296500030164...
0 )!2(
)1(
k
kk
k
B
... 243
518
2 2 1
60
k
k
k
kk
( )
3 .59048052361289410146... 2
9
.590574872831670752346... 6G
1 .590636854637329063382...
112
01 )!2(
2
)!()!1(!
1)2(
kkk k
k
k
k
k
k
kkI LY 6.113
...
5
1
4 .59117429878527614807... 2 2 1 4 2
0
2
arcsinh x dx
... log log( )1 2
21
12
0
1
2
xdx
x GR 4.295.38
= ( s GR 4.226.2in )/
1 2
0
2
x dx
...
1
3
!5
k k
ke J161
... 1
5 22 2
12
0
1
log sin( log )li
xx dx GR 6.213.2
... 5
...
5
6
9 .5916630466254390832... 92 ... 3log6
.59172614725621914047... dx
e ex 2
0
x 2 erndt Ch. 28, Eq. 21.4
1 .59178884564103357816... 21
2
1
2120
e erfk k
k
( )!
10 .5919532755215206278...
1
22
)!2(1cosh
2sinh2
k
k
k
11 .5919532755215206278...
0
2
)!2(2coscosh
k
k
k
eei
AS 4.5.63
=
1
2)12(
41
k k J1079
...
6
743234)3(182 )2(3
...
6
134)3(182 )2(3
...
01
4/1 sinh)!2(
!
2
1
2
2
dxxek
kerfe x
k
.592394075923426897354... 12
2
( )
( )
k
kk
...
1 2
sin
)4/11cos1(2
1sin
1cos45
1sin2
kk
k GR 1.447.1
... 2
)223log(
2
2log
22
22log
2
2
2
2log
=
1
12
2kkkH
...
12 3/1
1
322
3
k k
... log
( )( )
2
21 1 1
x
x xdx
...
12
)3(
2kk
k
k
H
...
1
3
2kkkH
... 231 e
... ii ee
i
1sin2
1
... csch
2 1
4 2112
1
2
1( )k
k k k
...
1 2
)1(13log2log1
kk
k
k
...
2
35
6
33
2
35
6
331
iiii
=
1
)13()13(k
kk
...
0 2)!2(
164
kkk
e
...
1 2
)1(44
kk
kpfe
...
1
0
1!
)(
k k
k
...
4
5
...
1
)2(log)(k
kk
... 2
4
... )!12(
33)1(
4
11sin
12
1
13
k
k
k
k GR 1.412.3
.595886193680811429201...
22 )1(
)3(23k
k
kk
H
.596111293068951500191...
H 1(
1
3 .59627499972915819809... log
...
e Eik
kk
( )( )
!1
1
0
=dx
e x
x dx
e xx x( ) ( )
1 10
2
0
=dx
x xe xx
21 01
1( log )
log( )
dx
... 4 11
4
0
e Eix dx
e xx( )( )
.5963475204797203166... ( )
1
2 1
1
1
k
kk
k
...
13)1(
log
4
1
2
2logdx
x
xx
1 .59688720677545620285... 11
22
1
kk
...
11
1)1(
kk
k
k
.59713559316352851048...
043 )1(3243
80
x
dx
...
1
2
)!2(
2
4
5
k
k
k
e
...
0
2
)3(!
2
4
1
k
k
kk
e
...
1 !
)22()2(
k k
kk
...
22/1
2
111)(
kk kkLikk
...
1
44 4cos2log
2
1)2sin2(log
k
ii
k
kee
...
0 )12(!
2)1(2
22
1
k
kk
kkeri
...
0
4
!
4
k
k
ke
1 .598313166927325339078...
2
324
1)2(128
127
48120 k
kk
...
1
2
4
3,1
224
kk
kk
G
.599002264993457570659... 2
1 1 1 1 2 1 11
12 2
2
1
cos sin sin log( sin ) sinsin ( )
( )
k
k kk
...
1 !
1
k kFk
... 1)3(log
... 3
)3(1
... log ( ) 21
kk
.5996203229953586595... 2 11
1 11
2
1
1
e Eik k
k
k kk k
( )( )! ( )!( )
=1
1 22 ( )!k k!k
= H
k k k kk
k k!( ) ( )!
2
1
11 1
1 .599771198411726589334... 2
11
k
kk k( )
...
1
24
4
)4(17
360
17
k
k
k
H
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
... arccsch2
= 3
5 42 1
1
F kk
k
.600053813264123766513... 3
2
5
24
2
22
1
22
1
2
2 2
2 2
logLi
iLi
i
= log( )
( )
1
1
2
2 20
x
x xdx
.600281176062325408286... 3 6 2 91
1 3
1
1
log( )
( /
k
k k k )
.6004824307923120424...
1
)22()2(
k k
kk
... e k
k
1
1
1/ !
.601028451579797142699... ( )
sinh
log
sinh(log )
log ( )
sinh(log( ))
3
2
1
1
2 2
0
1
0
2
1
0
x dx
e x
x dx
x
x dx
xx
.60178128264873772913...
22
12
1 111)1()1(
kk
k
kLi
kk
k
... 1
1 12 k
k
kk
log
... K1 1( )
... ( )33
5
... log10 4
8 .6023252670426267717... 74
2 .60243495809669391311... k
k kk k
2
21
21
2
11
1
1
...
2 5
51
10
1
521
coth
kk
... 24 5 50 32
3
5
21 1
4 24
2
) ( ) ( )
k kk
= 16 11 5 1
1
4 3 2
52
k k k k
k kk
( )
... 22
sinh
Berndt 7.2.5
1 .60343114675605016067... 2 2 21
23 arctan log
= log( )
12
1 20
xdx
2 .603611904599514233302... 11
1
k k
k
... log
!
k
kk
2
.60393978817538095427... 1
2
1
2 20
2
ee xx
/
/
cos dx
...
1
1
12
)(
kk
k
.604599788078072616865...
3 3
1
3 2
1
3 11
k kk
=
3 3
2
2 1 41
( )!!
( )!!
k
k kkk
= 1
3 1 3 20 k kk
)( )
= 1
21 k
kk
k
CFG F17
=dx
x x
dx
x x
x dx
x x2 2 2 4 2 120
20
4 20
1
= dx
x x
dx
x x20
202 4 3
3
=
03
06
3
06
123 811 x
dxx
x
dxx
x
dxx
xxx
dx
= dx
x x x x x5 4 3 20 1
= dx
e ex x
10
... log !8
.604898643421630370247...
1
1)1(
2
1,
2
1
2
121
k
k
k
... 10
... 7
48
3
83
9 3
16
1
4
2
3
2 21
log
log ( )
.605065933151773563528... 9
4 2
1
1
2 2
0
1
x x
xx dxlog
.605133652503344581744...
0 )1()1()1(
2
12
xe
dxEierfi
e x
... 9
8
3 2
4 2
2
22
log
( )
k
kkk 1
... sik k
k k
k
( )( )
( )!( )2
1 2
2 1 2 1
2 1
0
AS 5.2.14
.605509265700126543353... 3 2 4 4 ( ) ( )
... 112
2 2k kk
k log( )
... 13
... 5
256
3 43
31
sin( / )k
kk
GR 1.443.5
...
1
2)12(
21
2cos
k k
... log2
0
1
11
x
dx
... 4
11 2 2
0
ee xx cos dx
... dydxxy
yx
1
0
1
0
22
1
)log()3(3
... ( )( )
!/
12 1
11 1
2
2k k
k
k
ke
... 1 1
2
1
20 0ei
k ki
k
kk
k
k
/ ( )
!
( )
( )!!
... 5
.60669515241529176378... 1
2 1
1
2 2 1
2
211 1 2
kk
k kk
k
kk
( )
( )
1 .60669515241529176378... 1
2 1 2
2 1
2 2 11
0
1 12k
kk
k
k
k kk
k
( )
( )
=1
21
2
211 2jk
kj
k
kk
( )
Shown irrational by Erdös in J. Indian Math, Soc. (N.S.) 12 (1948) 63-66
... log 21
2 10
xe
edx
x
x GR 3.452.3
... 1
1 k kk ! ( )
.60715770584139372912... erf
e k
k
k
( ) ( )
( )
1 11
20
!
3 .6071833342396650534... 3 35
1024
15
1
( )
( )
( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
... 1536 5634
4
0
ek
k kk !( )
... 12 6 32 2 1
6 10
k
k kkk ( )
... 6 1
21
12 2
12
( )
( )k
k pk p prime
... 4
6081075
1
1
12
2 4 6 8 10 12
p p p p p pp prime
.60798640550036075618... 1
2
1
22
3
2
1
4 2 22 1 2 11
Fk
kkk
, , ,( )
1
=dx
x x2 22 2
... ( ) ( )2 5
... 3
2
3
2 31
log
Hkk
k
... 1
1 2 21 ( )! (k kkk
) Titchmarsh 14.32.1
... 2
31185
1
1
8
2 4 6
p p p pp prime8
.608699218043767368353... log logsinh
( )( )e e k
kk
k
4 2
12 11
1
K Ex. 124f
= log( ) ( )
log1
2 21
12
2
i i kk
... !
1)2(2
1 k
k
k
k
k
... Hk
kk
3
1 4
... log54
51
Li
... 2 2
3
1
3 40
4
sin
cos
/ x
xdx
... cosh( )
e e ee
ke e
k
k
1
2 2
2
0 !
...
2 4 3
302 15
9 31
( )( )
x
edxx
... log!
11
2
kk
.6106437294514793434... 2
3
G
... 8 2 e
= 11
18 3
1
3
1
33
21
24
H
k k k kk k
.611643938224066294941...
1
2 3
1
61 3
1 3
21 3
1 3
2
( ) ( )i
ii
i
= k
kk
k
k
k3
2
1
111 3 1
( ) ( ) 1
5 .6120463970207165486...
14
243 1
4 3
30
log x dx
x
1 .612273774471087972438... sinh
0
21
12
1i kk
...
12/3
1
2
14
2
3
k k
... 142
3 453 3 5
1
1
1 12 4
2 52
( ) ( )
( )k k kk
... 11
32
( )kk
... sinh2
21
42
1
kk
... 1
2 4
1
21
2 1
2 11
1
log tan ( )sin( )
k
k
k
k GR 1.442.3
... log6 3
3 .613387333659139458760... ( )
!
/k
k
e
kk
k
k
2
0
1
21
2 .613514090166737576533...
( )3
.613705638880109381166... 2 2 23
2
1
2
log hg
= 1
2 21 k kk
GR 0.234.8
= k
kkk 2 10 ( )
= ( )( )
( )
11
2
2 1
21 11
1 1
kk
k
k
kk
kk
= log x
xdx
1 20
1
= dxx )1log(1
0
2
= 4 1
1
1
2 1 220
1
0
1 log( )
( )
log( )
( / / )
x
xdx
x
xdx2
x
GR 4.291.14
= GR 4.384.10 log(sin ) sin/
20
2
x x d
4 .61370563888010938117... 6 2 22 1
3
1
log( )
k
kkk
128 .61370563888010938117... 130 2 22 1
5
1
log( )
k
kkk
.613835953026141526250... 11
11
2
1
3
1
321
tanh
k kk
.613955913179956659743... 2
5 5 5 2
7 10
50
/
dx
x
.61397358864975783997... 1
22 2log
... 4 42 1
1
1log ( ) /
k k
k
...
1 32
31
)1()!(
)!(
k kk
k
.614787613714727541536... 11
3 11
k kk ( )
... erfik kk
k
1
2
2 1
2 2 12 10
! ( )
=8
13
... arctancos
sin
/1
2 1 20
4
x dx
x
...
log(cos )( ) ( )
( )!1
1 2 2 1
2
1 2 1 22
1
k k kk
k
B
k k AS 4.3.72
7 .6157731058639082857... 58
...
1
2
1
2 2
1
22
1
2
1
22
1
2
21
22
1
2
cot
sin sin cos cos sinh sin
cos cos cosh sin
Adamchik-Srivastava 2.28 ( ) sin2 11
k kk
... arctan
!
k
kk
1
... 8 3 ( ) ... 2 4 4 4cosh e e
... 2 22 1
1
1 2
1
1log ( ) / / 2
k kk
k
k
k
...
1
2
2
10
2
)14(
4
4
)2(
1
)1(
16 kkk
k
kk
k
kkk
k
H
= x x
xdx
arctan
1 40
GR 4.531.8
... 2 3 6 2 31
12
130
log( )
( )( )
k
k k k
...
6
3
2
1
3
1
3 220
coth/
kk
... sincos
22
2
1
2
... 510
5 5
2 1 1
5 10
log( )
k
k kkk
... F
kk
kk 2 2
1
... 8
... k
kk
k
2
2 11 1
( )
...
3
7
2
3log3
324
15
...
0 05/12 316
5xx ee
dx
x
dx
...
11 5
2
1
... 5 1
2, the Golden ratio
... 1 2
... 1
21k
kk kH
... 4 2 22
20
log( )
k
kk
...
2 4
)(3)1(
4
2log9
8
31
kk
kk k
... 2
315
1
1
4
2 4
p pp prime
... 11
1 21
( )( )k kk
... 16
3
10 2
3
2 1
8 20
k
k kkk ( )
... 2 1
1
k
k k kk
... 16 6
1
3 121
cot
/kk 2
.620088807189567689156...
1
22
12
)2/1(
)1(2
2csch2
6 k
k
kk
... log( ) (
!
e k
k
k )
k
1
2
1 2 1
1
=
1 !
)12()1(
k
kkk
kk
B [Ramanujan] Berndt Ch. 5
= dx
e x10
1
... 4
9 3
1 3 2
1
e k
k kk
/
!
4 .620324229254256923750... 0
1 1
( )
( )
k
kk
!
.62047113489033260164... ( )( )
arctan
12 1
2 1
1 11
21
k
kk
k
k k
k
... 11
51
k kk ( )
... 1
31
4 2 3 6
22 2 62 3
1 31 3 5 6 1 3
2 3 1 3/
// / /
/ /( )
i
( ) /
/
/ / /1
3
4 2 3 6
2
2 3
1 3
1 3 5 6 1 3
i
= 6 3 1 16
613
2
k
k k
kk
k ( )
... 1
2
1
20, ,
... 2 2e /
... 2 2
2 224 22
2
2
1
2
1
2
log
logLi
iLi
i
= log( )
( )
1
1
2
20
1
x
x x
... 1
5 5loglog e
2 .621408383075861505699... 1
45 2 13 4 5 50 12 5 2 65 20 5
= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22
... )1(22
1cos1sin)1(1cos)1(1sin)1(1cos
2sicicici
= sin arctanx dx
x
x dx
ex10 0
= dx
e xx 20 1
.62182053074983197934... ( )
( )
1
2 12
k
kk k
... 2 3
6
CFG G1
... sinh sinh sinh32 4
311
43 3 1
1 3 3
8
e e e
e
6
= ( )( ( )
!
3 3 1 1
81
k k
k k
) J883
= 1
4
3 3
2 1
2 1
1
k
k k
( )!
.62322524014023051339...
02 1422
1arcsinh
xx
dx
... 1
21 2
1
21
1
2
1
2log arcsinh arctanh
= 11
4 1
1
4 11
( ) ( )k k
k k k
= 1
2 2 11k
k k( )
GR 1.513.1
= ( ) ( )!!
( )!!
1 2
2 10
k
k
k
k J129
= x dx
x x( )1 12 20
1
=
02 142 xx
dx
= sin
cos
/ x
xdx
2 20
2
= dx
x22 2
... 1
2 330 kk
... 21 3
2log
... pr k
kk
( )
2 11
= 13078
1251
21
6
1
( )k kk
k
k F
.624063589774511570688... 1
6 4 3
3
2
1
4 320
coth
kk
...
24 5 50 32
3
5
21 1
3 24
2
( ) ( ) ( ) ( )k
k
k k 1
= ( ) ( ) ( ) (4 2 4 11
24 1 4
1 1
k k kk k
)k
= 16 11 5 1
1
4 3 2
52
k k k k
k kk
( )
1 .624838898635177482811... K 2 1( ) ... ( )3
= 5
8
.62519689192003630759... 2
3 2
8 3
27 2
2
20
2
sin
( sin )
/ x
xdx
...
1 3
)2(
3csc
3log
kk k
k
1 .62535489744912819254... 11
22 21
kk
... 1
4
AS 6.1.10
... 2 1
12
2
2
2
k
k
k
k
k
ke
k
( )
!
...
2 2 1
2
0
sech
e
e
x
xdx
/ cos
cosh GR 3.981.3
=
dxee
xxx
cos
.626059428206128022755... ( )( )
12 1 11
21
22
k
k k
k
kLi
k
... 22
3 2 21
16 21 31
( )log
( )/
x
x xdx
.626503677833347983808... 6 1
5 4 1 221
sin
( cos )
sin
k kk
k
.626534929111853784164...
( ) ( )2 21
k kk
1
... S k k
k
k
kk
22
1
2 2
2
( , )
( )
... 71 4/
... 1
2 2
... sinh( )! (
22
2
2 1
4
2
2 2 2 1
0 0
e e
k
k
k
k
k
k
k )!
dx
AS 4.5.62
= e GR 3.915.1 xx2
0
cos sin
.6269531324505805969...
( )
( )
3
2 1
1
213 3
1
kk
k k
k
.627027495541456760072... 3 3 4 2 31
3 221
log log/
k kk
.627591004863777296758... ( ) )2 6
... 2
3
1
3
2
3
1
3
2
3
321
,k
k k
kk
= dx
x x
x
xdx2
3
301
1
log( )
= dx
x10
2
cos
= e
edx
x
x
/3
1
= log log( )
1 14
23
0 0
x dxx x
dx
.627716860485152532642... ( )( )
log
13 1 1
111
1 23
k
k k
k
kk
k
... 1
42 1
1
2 10
e erfik kk
!( )
.6278364236143983844... 4
5
4 5
25
1
2 arcsinh
= 2 10
111
2
1
4
12
Fk
k
k
k
, , ,( )
... sin( )cos log( sin ) sin
1
22 1 4 1 12
= sin
( )
2
1 1
k
k kk
... 5 1
3 2
50
x dx
x
/
... sin ( ) sin/
1
2
3
21 3 i
= cosh( )
3
21
3
2 1 20
kk
1 .628473712901584447056... 1
11 k k
k
... Lik
k
k3 3
1
( )
... ( ) cot ( ) cot ( )/ / / 14
1 11
21 4 3 4 1 4
i
= ( )( ) ( )
// /
1
42 1 2 1
1 43 4 3 4
( )
( ) ( )/
/ /1
42 1 2 1
3 41 4 1 4
= ( ) ( ) Im( )
1 4 2 11
21
1
2
42 1
k
k kk
k
kk
k
k
i
14 .628784140560320753162... 2 35
64 1 1
9 11
1
22
2
2
32
( ) ( ) ( )( )
k kk k
k kk k
... ( )/
1 11
41
kk
k
e
k
... 0
1 112
1
22 2
( )kk
kk j
kj
2
... 24 21 164 2
0
dx
x( / )
... 2 1 2 112
1
e eEik k
kk
( )( )
!
1 .6293779129886049518...
2 12
22
2
2 2
20
loglog log xdx
e x
2
= 2 2
2 2
012
2
12
( log )
loge xx dx
1 .629629629629629629629 = 44
27 4
3
1
kk
k
... G
... 1
22
12
log( )
( )
k
k kk
= ( ) log2 2 11
11
21
kk kk
... 11
2
p
p prime
.6304246388313639332... 11
21
362 4
( )
... ( )34
7
... 2
e
... log3 2
... 3 1
4
3
3
3
0
sin sin
k
kk
Berndt ch. 31
... 1
2 11k
k k ( )
=12
19
... 105
16
9
2
... e dxxsin
0
1
.6319660112501051518... 9
4
1
3 21
F kk
GKP p. 302
.631966197838167906662...
1
2
2
22log
12)3(
kk
k
k
H Berndt 9.12.1
.632120558828557678405... 11 1
0
1
31
ex x dx
x
dx
xcosh cosh
.632120558828557678405... 11 1 1
11
21 0
1
e k xx e x d
k e
x( )
!( ) log x GR 4.351.1
=
13
1
0
1coshcosh
x
dx
xdxxx
... 22
.6328153188403884421... ( )
logk
k kkk k
1
3
11
1
31 1 k
1096 .633158428458599263720... e7
6 .63324958071079969823... 44
... cos( )e
... 3
1 3
... 1
11
0
( )k k
k
2 .634415941277498655611...
loglog sin1
2 1
2 2
20
e x
xdx
... Ei ee
k k
k
k
( )!
11
...
1
2
1
4/1
)1(
2csch2
k
k
k
... arcsec15 33
6
Associated with a sailing curve. Mel1 p.153
...
2
1
2
2 2212
13
161 2 1
2 1
1
( ) ( )( )
k kk
kk k
.635181422730739085012... )()1(22
2log
1
kk
k
=
xx
dx e x dxxlog log log1
0
1
2
0
GR 4.325.8
... 2
12
142
1
4
4
4 1
Gk k
kkk k
( )
( )
...
2 )(
)2(1
k k
k
.63575432737726430215... 2 2 2 31
42
1 32
( ) ( )( )
k
kk
= ( ) ( )( ( ) )
1 12
k 1k
k k k
... I k
k kk
1
21
2
2 2
( !)
... 1
2 2
1
120
csch
( )k
k k
= 7
11 63 1
1
F kk
k
... 2 1
1 222
0
logsinx x dx
... 2
3112 16
1
96
2
sin /k
kk
GR 1.443.5
= iLi e Li ei i
2 32
32 / /
... ( ) ( )2 7
... 2
11
4 21
kk
GR 1.431
= cos
2 11
kk
GR 1.439.1
= 0
1 2/
= i e ilog
... 1
5
1
51
1
43 5 2
5 5
24 5
( ) / i
1
51
1
43 5 2
5 5
21
1
43 5 2 5 51 5 2 5( ) ( ) (/ / i i
( )(
/1
5
1
43 5 2 5 5
1 5
i
= ( ) ( )
1 5 3 11
1
1
3
52
k
k k
kk
k
...
2 2 2
1
2
1
2 11
2
221
1
1
coth ( )( )
k
k
k
kk
k
... 6 1
4 2 5 2
2 2
1
sin
cos
sin
kk
k
... 3
2
2
12 2
e
x
xdx
cos
( )2
... 1
2 422
4
0
cosh( )
k
k k 2 !
... ( ) ( )k k
kk
1 1
2
1 .6382270745053706475... 1
2 dkk
... 26
442
21
cos k
kk
GR 1.443.3
.6387045287798183656... 3 9 3
24
1
6 421
log
k kk
= dxx
x
1
05
6 )1log(
.638961276313634801150... sin log Im2 2 i
... 1
3
5
3
5
3 1
4
0
3
x dx
ex
... i
i i
i i
k
kk
k2
21
22
1
2
21
22
1
2
14 2
2 11
1
log ( )( )
1
= arctan1
22 kk
.639446070022506040443... ( ) ( ) ( )3 4 2
.6397766840608015768... k
kk 3 11
... 3
8 83 3 43
43i
Li e Li ei
Li e Li ei i i( ) ( ( ) ( ) i
= sin3
41
k
kk
... 4
32 log
1 .640186152773388023916... 7
32 log
.64031785796091597380... dxx
xx
1
0
2
1
log
27
502)3(16
... H
kk
kk
( )
( )
3
1 2 2 1
... 1
2 2 2 1 2
21 3 1 3 2 3 1 3
3
32 ( ) ( )/ / / /
k
kk
... 22
e
... H ( )/
31 3
.641274915080932047772...
2 6 2 3 320
20
dx
x
dx
x 2
... 21
3 33
2
0
1
Lix
xdx
log
... 3
24
5
5
6
1
6
hg hg
1 .641632560655153866294... 1
2
1 1 2
1 1 21 2
2
2 111
k k
k
kkk
( ) /
/
/
[Gauss] I Berndt 303
... 10
... ( )
1
2
1
1
k
kk k
... ( )( )
( )!sin
14 2 1
2 1
11
12
2
k
k k
k
k k
.642092615934330703006...
122 1
1211cot
k k GKP eq. 6.88
=
0
2
)!2(
4)1(
k
kk
k
k
B
... cot1
... 7
G
.64276126883997887911...
4
32Li
.6428571428571428571 = 9
14
1 .64322179613925445451... 11
32 6
4 2 3 6
21 3
1 3 5 6 1 3
( )/
/ / /i
1
3
4 2 3 6
2
1 3 5 6 1 3
i / / /
= 6 3 1 16
613
2
k
k k
kk k
( )( )
.643318387340724531847... ( )
/
k
kLi
k kk k
1 1
21 2
2
1
.643501108793284386803... 2 22
2arctan log
gd
9 .6436507609929549958... 93
... e
x xesinlog sin log
1 1
2 1
dx
... e ee
kk
k
k
sin ( )( )
12 1
1
1 !
... Gi
Lii
Lii
log2
8 2
1
2
1
22 2
= GR 4.227.11 log( tan )/
10
4
x dx
=
log( )1
1 20
1 x
xdx GR 4.291.10
= arctan
( )
x
x xdx
10
1
GR 4.531.3
.643796887509858981341...
( )1 e
eHe
.643907357064919657769... 1
6 4 2
7
2
1
4 421
tan
k kk 6
...
0
2
3 2
sin
2log42log
1x
x
... 2 22
( )k
k
... log jj
kjk
2 112
... G5
... 11
41
k kk ( )
.6449340668482264365...
2
61 2 ( )
...
2
2)1(
2 1)2()2(1
6 k k
J272, J348
=
12
12 )1(
1
1
1
kk kkkk
= ( ) ( ) ( ) ( )k k k kk k
1 2 22 1
2
2 1 = ( ) ( ) ( ) ( )
1 12 1
k
k k
k k k k k
= 1
21
21k
dx
xk
=
x x
xdx
log
10
1
GR 4.231.3
= log x
x xdx3 2
1
= 1
1 1 20
1
x
x x
xdx
log
( ) GR 4.236.2
= x dx
e ex x
10
GR 3.411.9
...
2
12
12
162 1
1 2 1
1
( ) ( )/
( )( )
k
k
kk k
= H
k kk
k ( )
11
= 31
221 k
kk
k
CFG F17, K156
= k k k k kk k
( ) ( ) ( )
1 1 11 2
=
log x
xdx
10
1
GR 4.231.2
=
0 )1(
)1log(
xx
x
= log
( )
log
( )
log
( )
log
( )
2
20
1 2
21
2
30 11 1 1
x
xdx
x
x
x
xdx
x
x xdx
1
= log( )1 2 2
0
1
x x
xdx GR 4.296.1
= GR 4.223.2
log 10
e dxx
=
0 1dx
e
xx
=dx
e x
10
...
2
261 1
k kk
( )
...
2
262 1
k k kk
( ) ( ) 2
.64527561023483500704... 3
62 2
3 3
2 log
log
= log( )( )
11
1 20
x xdx
.64572471826697712891...
1
)(
k kF
k
... 7
... 3
31
1
31
sinh
kk
.645903590137755193632...
0
2
1
sin)21()21(
4
1
2 xe
xii
.64593247422930033624...
2
)1(cot
2
)1(coth2)1(coth25
16
1 iii
=
8
1
16
1
81 2 1 1 2 13 4 1 4 3 4 1 4coth ( ) ( ) ( ) ( )/ / / /
1
81 2 1 1 2 11 4 3 4 1 4 3 4( ) ( ) ( ) ( )/ / / /
=
1262
1)68(1
kk
kkk
... 11
2 2
.646458473588902984359...
1
33
18
3 4
3
2 2
3
1
3
1
42
2 2
2Li Li( )log log log
= x
edxx
30
.646596291662721938667... 11
25
24
25 5
1
21
1
21
1
arcsinh ( )!( )!
( )!k
k
k k
k
... 1
45 1 3 1 1
2 2
2
1
sin cos ( )( )
k
k
k
k !
... ( )
log( )6 4 1
11
4 6
2
k
kk k
k k
.64704901239611528732... 1
2
1 3
22 11
1
1k k
k 1
k
k
kk
( ) ( )
= 11
17
3 .647562611124159771980... 36 6
22
( )
...
0
2cosh)1(4
2
dxxee x
... 1
2
1
2
1
21
1
2 21
coshcos
sinhcos
sinsin sin
!
k k
k kk
.647793574696319037017...
dxx
x
1
3
arcsinh
2
21log
2
1
2
1
... cos3
2
... 3 3
2
22
1
log log( )
x
xdx
... sech12 1
212
0
e e i
E
kk
kcos ( )! AS 4.5.66
.648231038121442835697... 3
5
2
5
5
20
2
0
arctan log log cosx x
edxx
... ii eLieLi 22
2
2
1
3
1 .648721270700128146849... ek k
ikk k
i
1
2
1
20 0! ( )!!/
...
2
15
2
15log
4
3
12 22
2
Berndt Ch. 9
.64887655168558007488...
1 )24(
)4(
k k
k
... 4
1
2
3tanh
32
= k
kk
k k
4
62 11
6 4 1
( )
...
0 )2(3
13
2
3log9
kk k
... ( )
log( )5 3 1
11
3 5
2
k
kk k
k k
... 4/
04cos6
1
3
2log
3
x
dxx
... 2
1sin
= 20
13
1 .650015797211168549834...
1 !)1(
1log
k
kk
ke
kk
Be
e
e Berndt 5.8.5
...
1
2
)6(!15894320
k kk
ke
.650174578577121012503...
1
2
)6(!15894320
k kk
ke
...
2
3 23
41
3
4 31 1
log ( )( )k
k
k
kk k
...
3/1
3/23/2
3/1
3/13/1
3/2
2
)1(1log)1(
2
)1(1log)1(
3
2
3/1
3/2
2
11log
3
2
=
0 )23(2
1
3
2,1,
2
1
3
1
kk k
.6503861069291288806... iLi e Li e Li e
k
ki i i k
k212 2 2
12
1
( ) ( ) Im ( ) ( )sin
1 .650425758797542876025...
0 )12(!
121
k kkerfi
.650604594983248538006... 1
21kH
kk
...
1
3 )12(2)12(27
)1(412
2 k
k
kk
=
2/
02
22/
02
2
sin1
cos
cos1
sin
dxx
xdx
x
x
= 1
0
22
2
)arctan(1
dxxx
x GR 4.538.2
...
22
1
1arctanh
)12(
1)24(sinhlog
2
1
kk kk
k
2 .6513166081688198157...
1
2
3
428 3
12 33
43
0
( ) ( )( )kk
.651473539678535823016... 15
32
5
4
35
64 5
2
1
log
k Hkk
k
... 2log
.651585210821386425593...
2
2 2
2 21
4 2
29 4
1 4 1 4
1 4 1 41
1/
/ /
/ /
sin sinh
cos cosh( )
( )
kk
k
k
= 1
2 2 21 k kk
...
22
2 )(
k k
k
... 31
12601
6
0
1 log( )
logx
x
xdx
... )7()2(
...
1
34
34
3
2 4sin
23
16
3
24
k
ii
k
keLieLi
i
GR 1.443.5
...
2
2 1)2(k
k
...
1
3
!!2
1
211711
k k
kerf
ee
...
2
55
24
59
2
552514
20
1 ii
2
55
24
59
2
55251
20
1 ii
2
55
24
59
2
55251
20
1 ii
2
55
24
59
2
55251
20
1 ii
=
1232
1)35(1
kk
kkk
.653743338243228533787... ( )( )!
15 3
2
1
212 3
1
kk
k k
k
k k
... ( )
log( )4 2 1
11
2 4
2
k
kk k
k k
1 .654028242523005382213... coshcos sinhcos sin( sin )sin
!1 1 1 1
1
k k
kk
... 2 37
41 1 1
2 3 3 1
11
3
2
2 32
( ) ( ) ( ) ( )( )
( )
k
k k
k k kk k
k k
... /1e
... ( )3 1 1 1
21
2 22
k
kk Li
kk k
... 3
7
1
3
2
0
( )k
kk
k
k
... 11
61
k kk ( )
... )3(
8
... 163 1 1
42
1
Gk
kk
kk
( )( )( ) Adamchik (27)
.655831600867491587281...
1
1 !)1(
kk
k
k
k
... ( )2
2 1
c Patterson Ex. A.4.2
... ( )2 1 1
31
3 22
k
kLi
kk k
...
5
33Li
...
2
/12
2
2
)1(1
!
1)(
k
k
k
kekkk
kk
...
1
22
1
2
1
2
1coth
k k J951
... 4 2
3
2 3
31
42 1
1
log log
H kk
k
... 5log6
... dxx
x
1
06
2
1
)1(log
32
32log
2
32log34
1 .65685424949238019521...
0 8
112424
kkk
k
5 .65685424949238019521... 32 4 2
... ( ) logk k
kk 22
...
1
21
2
i i
...
2
3
2
3
22592
5533
4 iLi
iLi
i
=
13
6/sin
k k
k GR 1.443.5
... )3()4()2(
)4(
15
2
=
primep kk
k
primepk
k
pp
p
k 022
2
12
)( )1(
1
)1(
HW Thm. 299
=
12
)(
k k
k HW Thm. 300
1 .6583741831710831673...
22
22
062 2 2
2
log loglog ( )x dx
ex
.65847232569963413649... 2 2
20
12
4
7 3
8 1
log ( ) arccos
x x
xdx
= dxx
x1
0
2arcsin
... 45
4)4(8
4
... dxx
xex x
2
2/
0
tan
cos
4sin
2
3 2
GR 3.963.3
.658759815490980624984...
24
14
111)1(
kk kLi
kk
k
.658951075727201947430... 1
21
1
21
Eik kk
( )!
... 4
3172
31
( )( )k
kk
... )1sinh(sinh)1cosh(cosh1)2sinh2(cosh eeee
=
02 )2(!
1)1(
k
ke
kk
e
e
ee
...
2 1
/1 11
!
)()1(
k k
kk
ke
k
k
...
k
k
k
k
8
)1(2232
3
240
8 .6602540378443864676... 75 5 3
.660349361999116217163... 11
3 121
k kk
...
1
2
2
22
2 )2/(sin
4
1
128
1
4 k
ii
k
keLieLi
... 4
21
8
1
421
coth
kk
J124
=
0 1
cossinxe
dxxx
...
044/9 22 x
dx
...
05
2
15
3csc
555
2
5
2dx
x
x
... G
...
22 1
log
k kk
k
...
1 !
)1(
k
kk
k
kB
...
0 !
2cosh2sincosh2coshsinh2coshcosh
k k
k
... log cos log ( )( )/ /21
4
1
21 12 2
e ei i
= ( )cos /
121
1
k
k
k
k
... 7)32log(42)21log(21362174105
1
avg. distance between points in the unit cube
... 1
4
3 3
8
23
1
log log( )x
xdx
... )6()2(
...
1 )!(
1
k k
...
1
3
k
k
k
...
4
1
8 14 2 1
2
42 1
coth ( )
k
kk
k k
...
1 3
1
k kF
...
1333
4/cos
3
1
3
1
2
1
k k
kiLi
iLi
... )3(
2
... 41
12
120
Gk k
k
k
( )
( )( )
...
13
41
)2(3
)(
164
4
1
2
164)3(28
k k
...
03
41
)2(3
)(
1
4
1
2
1)3(28
k k
=
1 )1(4
12
140
93
kk
kk
4 .66453259190400037192...
0
2)12(
21
2cosh
k k
... 0 1 0 20
11 2 1
Fe
Je k e
k
kk
; ;( )
( !)
.66467019408956851024... 3
84
0
2
x e dxx
= dxxx
xe x
2/
06
2tan
cotcos
cos212
GR 3.964.3
.66488023368106598117...
2
061 1 2
1
( ) ( )logx x
e edx
x x
... ( )
log( )3 1 1
11
3
2
k
kk k
k k
... 2 2
1 .66535148216552671854...
5
212
55
55log55log5
10
3
5
6
=
2 5
1)(6
kk
k k
...
1 2k
k
kk
kH
... 28
...
arctanh1
1 e
.665738986368222001365...
12
2 )(
k k
k
... 4
arctan
... 25
...
20
12
222
)!(
1)2(
kk kkI
kk
k
...
0 22
)21arctan(2
24
3xx ee
dx
...
23
3 )(
k k
k
...
12
/2
0 !
)2(2
k
k
k
k
k
e
k
k
... 22 ... 1sincos ... sin ( ) cos ( ) cos1 1 1 1si ci 1
= log sin( )x x10
1
dx
.666666666666666666666 =
k
k
2
)1(
3
2
=
1
1
0 )2/1)(2/3(
)1(
)2)(1(4
12
k
k
kk kkkkk
k
=
1
34
8kkkF
=
1
2)2(
11
k k
=
23 1
21
k k J1034, Mardsen p. 358
=
2/
02sin1
1
dxx
= dxe
xx
02
cosh
= cosh log xdx
x31
=
0
8
3xe
dxx
1 .666666666666666666666 = 3
5
=
1
14
8kkkF
2 .666666666666666666666 = 3
8
=
1
4
8kkkF
...
0 )!!!(
1
k k
.666822076192281325682... 21
...
0 )!12(
3
32
3sinh
2
3cosh
k
k
k
k
.66724523794284173732... 21 2
2
1 1 12
1
log( ) ( )!
!
k
k
k
k k
... 4
arccos
...
2
1
42 22 2 e ei i
( ) sinkk
1 11
2 k
...
0 12
)1(
4
3,
2
1
4
1,
2
1
2
1
k
k
... ( )2 1 1
21
2 22
k
kLi
kk k
...
2
1
6
2log
6
2log)2( 3
32
3 LiLi
=
0
2
13
21
0
2
22
log8
21
logxe
dxx
xx
dxx
x
dxx
...
122
)(
kko
k
k
...
1 )1(12
1
kk k
...
0
323
1)3(122log82log2
xe
dxx
1 .6701907046196043386... k
kk 2 11
...
1
1
kke
... 55
... 3
2 3
...
0 )2(
111log1
kk kee
ee
.671204652601083... ( )
/
12 3
2
k
k k k 2
...
15
/11)1(
k
kk
k
e
1 .67158702286436895171... 2
12sin
!
11cossinh1coscosh1sin
2
1sin
0
k
kk
... e erf e
xdx
x ( )1 1
2 1
2
20
...
1
021
1loglog2log
6
5
6
1log
3
2
xx
dx
x
GR 4.325.6
=
01 1
log
ee
xx
.67177754230896957429...
033 )1(327
10
x
dx
...
13
1)()(
2
1
k kkii
=
1 1
1 )14()14(2
11)12()1(
2
1
k k
k kkk
=
12 )(
1Re
k ikk
.672148922218710419133...
1 )2/5(
1
5
2log4
75
92
k kk
...
12 24
1
2cot
244
1
k k
...
0
13
13
)13()1(
3
31log
3
1
34 k
kk
k
...
2
3 1)(2)(k
kk
...
2
3 )()(k
kk
...
2
3 1)(k
k
... 2
2cos
4
1
4)1cosh()1cos(
2
2cosh2cos2
2
e
eii
=
0 )!4(
16
k
k
k
.6731729316883003802... 1
2
3
4 83 4
2 4
20
/ cossin
x
xdx
... 1sinh
21
ee
.67344457931634517503...
1 )(
1
k kq
... 1
2 2
1
2
1
2
1
21
2
21
HypPFQx
xdx, , ,
sin
.6736020436407296815... kk
k
2
1 4 1
... 2
15log
8
3
20
2
Berndt 9.8
...
kLi
kk
k
kk
11
)1(
1)(
13
23
...
1
3
!8
))1(1)(33(1cosh
k
kk
k J844
...
1
31
22
)12(
)12(2
2
)1(6
42
)3(7
kkk k
kkk
3 .674643966011328778996... pk
kk 21
the kth prime pk
1 .674707331811177969904...
2 )1)((2
11
kk k
...
1 2
1)2(
k kF
k
...
22)1(
log
k k
k
... 1
pp prime !
...
0 )6/1)(1(
)1()32log(32log
5
6
k
k
kk
...
0
2 2
sin
2log1
1x
x
.67551085885603996302...
02 12322
2
1arctan2
2
2arctan
xx
dx
= dx
x x20 2 3
14 .67591306671469341644...
1 1 1
3
222
1
02
)21(
42
22
)(
n m mm
kk
mnk
k
knmkk
.675947298512314616619...
2/1
04
4/
04 1
1
sin1
cos
24
5log2arctan
22
1dx
xdx
x
x
...
0
2
)!12(2
0sinh
k
k
k
ee
i
=
12
11
k k
...
21
1arctan
1
12
1)2(
kk kkk
k
... ( )
!
k
kk
1
... ( )k
kkk 21
...
10
8/1
!)!2(
!)!1(
2!!
1
22
1
21
kkk k
k
kerfe
... ( ) ( )( )k kk
k
2
.677365284338832383751...
0 )2()!2(
11sinh141cosh1812
16212
k kkee
... 3
2 12
1
21
2
2
( ) ( )k k
k
... )()()(3)(38
33
3333
iiii eLieLieLieLii
=
13
3sin
k k
k
3 .678581614644620922501...
2
2
)!1(
)(
k k
k
17 .67887848183809818913... e erfk
k k
k
k
2
1
2 22 2
2 1
( )!!
( )!! !
... 1
3
... 2
1cot1
... 4
11
4 22
12 1 2
2
20
eK
x dx
x/ ( )sin
= GR 3.716.9 2/
0
2 )(tansin
dxx
...
1
02
6
77
7
1
log)()(360
256
61dx
x
xiLiiLii
GR 4.265
... 1
2 112j
kjk
...
2
22 1)(k
k
...
03
01
2
)2(!4
1
)!2(4 xe
dx
kk
kex
kk
...
2
1)(
k
k
k
kH
0 =
1
22
425
292
kk
kFk
...
22
1)(1)1(
)()1(
kk
k
k
kHkk
k
... 1
log1
1)(22
k
k
k
kkH
kkk
7 .6811457478686081758... 59
.68117691028158186735... 1
2 2
1
121
cot
kk
...
12
1
x
dx
... 4/18
... 3
2111 1
1
411 2 1
2 12 1 2 1
2
22
F ii
F ik
kkk
, , , , , ,( )
... 4/
0
2232
cot192164
2log dxxxG
...
1 11
0
1 3
1
3
)(
13
1
j kkj
kk
kk
k
=
1 )13(3
132
kkk
k
Berndt 6.1.4
...
1 !
11
k k
5 .68219697698347550546... dxe
xdx
xx
xdx
x
xx
1
3
12
31
02
34
1
log
1
log
4
)4(21
120
7
GR 4.262.1
.682245131127166286554... 11
22 2 2 2 2 2 1 1
1
k
k
k( )
... )2/(sinh2
... 3log5
... 11
71
k kk ( )
... 3/1
...
1
2 1)1(k
k
...
0
2)12(2
11
22cosh
k k
... )(1sin2 ii eei
... coshcosh sinhcosh sinhsinh1
2
1
2
1
2
J712
= 1
2
21
0
e ek
ke e
k
/ sinh( / )
!
.683150338229591228672...
12
)1(22
33
2
2
13log
8
93log
4
3
24
7
k
k
kk
H
...
1
1sin1log
1
k kk
...
1
0
2
1
1
!
22log)2( dx
x
e
kkEi
x
k
k
.683899300784123200209... ( )k
kk 21
... 2
22
12 2
162 2
2 1
2
2
4
log( )!!
( )!
( )!
( !)
k
k k
k
k kkk
k
=H
k kk
k ( )( )
1 2 11
=log ( )2
20
1 1
x
xdx
= 19
13
.6842280752259962142...
( )
( )
k
kk
1
22
21 .68464289811076878215... dxx
x
1
04
442
)1(
log)3(9
90
7
3
... log log( )
( )Kk
k jj
j
k
k0
1
1
2 1
1
22 1
11
K Khintchine's constant 0
= log ( )22 2
2
21
2
1
21
4
Li Lik
k
k2
= ( ) log21
22
1
12
2 22
Likk
=
kkk
11log
11log
2
Bailey, Borwein, Crandall, Math. Comp. 66, 217 (1997) 417-431
.68497723153155817271...
112 1
)(
)1( kk
kk
kk
... 11
2
2
k k
k
( )
log
Khintchine's constant
1 .685750354812596042871...
4
1K
...
2
11)(k
kk
...
2
3531
2
35312
6
1 ii
ii
=
1
1
13
1)3()1(2
1
1
1
k
k
k
kk
=
1212
1)13(1
kk
kkk
.686827337720053882161...
12 2
1
2
1
2
7tanh
7 k kk
.686889658948339591657... 7
)3(4
.6869741956329142613... 3
4
G
...
22 3
1)(
k k
k
... )4()2(3
... 8
32
3
2
3
tanh tanhsin
x
e edxx x
... ( )
cos( )/
2 2
29 42
0
2
e xx dx
.6885375371203397155... 11
41
kk
...
02/3
2/3
)12(
1
2
321
k k
...
02 )!2(!
1)2(
k kkI LY 6.114
...
2
/1
1
1!
1)2( 2
k
k
k
ek
k
3 .689357624731389088942... (log )log( )
2 25
1
2
20
arccschx
xdx
...
0
22
3
12log22log2
6 x
x
e
ex
...
2 5
)(4)1(
55
55log
5
15log
5
21
5
21
kk
kk k
... 2
arccsc
.69019422352157148739...
2 1 40
2
ee xx
/ cos
dx
7 .690286020676767839767... 3log7 4 .69041575982342955457... 22
... cosh
sinh cosh2
0
11
2
1
2 x x dx
...
0 )2(6
12
3
43
kk kk
k
... 1
22 2 3
3
1
log coscos
k
kk
... 1
21 1
1
2 1
1
1
(cos sin )( )
( )
k
k
k
k !
...
02
12 2/12
1
2/1
1
2cot
21
kk kkk
=
112
)2(
kk
k
...
211
2
0
2
2
0
2
J x x dx x x dx( ) cos(sin )cos cos(sin ) sin/ /
= sin(sin ) sin/
x x d0
2
x
...
1
4
481
380
kk
k
... 12 102
1
31
I
e
k
k
k
k
( ) ( ) ( )!
( !)
2
.691849430120897679...
sin( ) sinh( )
2 2
61
42 4
2 k
.691893583207303682497...
1
2
)2(22log42log24
kk
k
k
kH
... ( )
( )
2 1
21
k
kk
.692200627555346353865...
0
/1
!
)1(
kk
ke
eke
= 13
92
.6926058146742493275... 1
2 22 k kk log
...
1
1)1()1(2log
k
k
k J71
=
2
1
2
11
1
Likk
k J117
=
0
12 )12(3
12
3
1arctanh2
22
1arcsinh2
kk k
K148
=
2 2
)(
kk
k
=( )
log( )2 1
11
2
2
k
kk
k k
K Ex. 124(d)
=
112
112 42
)2()14(
2
)12()12(
kk
k
kk k
kkk
=
0 )22)(12(
1
k kk GR 8.373.1
=
11 1 )2(!)!2(
!)!12(
!)!2(
)!1(
)!2(
)!22(
kk k kk
k
k
k
k
k J628, K Ex. 108b
=
13 28
1
2
1
k kk [Ramanjuan] Berndt Ch. 2
=
1
3 28
)1(21
k
k
kk [Ramanujan] Berndt Ch. 2
=
1
3 327
)1(
2
3
4
3
k
k
kk [Ramanujan] Berndt Ch. 2
=
12 14k
k
k
H
=
1211
H nk
kkn
( )
=
1
cos
k k
k GR 1.448.2
= limn k
n
n k 1
1
[Ramanjuan] Berndt Ch. 2
=1
212 ( )s rsr
J1123
=
2 132 xx
dx
xx
dx
=dx
ex
10
=
1xe
x
e
dxe
=dx
x x
dx
x x2 3 1 320
20
1
= dxx
xdx
x
x
12
1
02 )1(
log
)1(
log GR 4.231.6
= dxx
x
122 )1(
log GR 4.234.1
= dxx
x
1
03
4 )1log(
=
4/
0
2/
0 cos)sin(cossin1
xxx
dxdx
x
x
=
03
4sindx
x
x
= GR 4.393.1 dxxxdxxx 2/
0
2/
0
tanlog)cos(tanlog)sin(
= GR 4.384.3 1
0
2cos)log(sin dxxx
=
02 )2/cosh()1( xx
dx
= dxxx
x
0 cosh
)2/tanh( GR 3.527.15
= GR 6.2111
0
)( dxxli
... 2 22 1
21
logk
kkk
=log( ) log2
21
x x
xdx
...
020 3)!(
)1(
3
2
kk
k
kJ
...
1
3
!
)(
k k
k
... )3(
...
23
2
2
)1(
21)()1()3(2
3 kk k
kkkk
... 4/34/34/14/1 )1(1)1(1)1(1)1(14
i
=
1
1
14
1)14()1(2
1
1 k
k
k
kk
k
...
1
4/1
)!2(
!
2
1
4
3
4
1
k k
kkerf
e
... /1
...
1
1
)!2(
4)1()2(2log2
k
kk
kkci
...
22 2
log
k kk
k
... 6 9 .695359714832658028149... 94
...
1
2
2
)1(2log668
kk
kk
... ie ei i
22 2
( ) sink kk
1 11
...
1
2
)!2(2sinh2
k
k
k
ee
...
0 )!2(
coshk
k
k
ee
... u k
kk
( )
2 11
... Ei
e
k ek
k
k
k
( ) ( ) ( )
!
1 1 11
1
...
1 2
1
kk k
...
2/1
2
3
16
... 21
3 33
2
21
Lix
x xdx
log
.69778275792395754630... dxxx
xLi
1
0
2
2
)1)(2(
log
2
1
3
1
18
... 11
2 122
kk
.698098558623443139815...
1
1
15
1)25()1(2
1
1
1
k
k
k
kk
7 .69834350925012958345... sinh
!( )
4
2
41
20
k
k k k
.6984559986366083598...
0 )!12(
2)1(
2
2sin
k
kk
k GR 1.411.1
=
0
1
)!2(
2)1(
k
kk
k
k
.699524263050534905034...
12 2/2
12log68
k kk
.699571673835753441499... | ( )| k
kk 2 11
=10
7
...
3/13/2 )1(44
2
1)1(1()1()1(
6
1 ii
3/13/213/23/1 )1(44
2
1)1(1)1(1)1(1
6
1
3/23/1 )1(1)1(16
1
=
1
1
133
1)36()1(2
11
k
k
k
kkk
.700170370211818344953...
122
12
11
)1(
)12(
kk kkLik
k
k
2 .700217494355001975743...
2
33 1)(k
kk
.700367730879139217733...
122
11
kk
.700946934031977688857...
1 )!2(
),2(2
k kk
kkS
1 .70143108440857635328... x dx
e xx
2
0
.7020569031595942854...
1
31
3
1
2
1)3(
x
dx
kk
.70240403097228566812... )()1()()1(4
1
2iiii
= )1()1(4
1
2
1ii
i
= 1
12 11 k k kk
.70248147310407263932...
02 552 x
dx
.702557458599743706303...
0 )2(2!
124
kk kk
e
.703156640645243187226... 2log23
8
2
5
1 .70321207674618230826...
2 12 69
4)(
3
23log
33 k k
k
kkk
...
0
2/)12( )12(
1
4
1tanhlog
2
11arctanh
kk kee
J945
...
4
1sin2log
))2/1cos(1(2
1log
2
1 GR 1.441.2
=
1
2/2/ )2/cos(1)(1(log
2
1
k
ii
k
kee
...
1
2!
22,2,2,2,1,1,1
k
k
kkHypPFQ
...
222 2
11log
2
)1(2
1)(
kk
kk kk
k
... 4G
.7041699604374744600... dx
xx1
... 2
4
...
( )
( ) ( )
( )
( )
k
k k
k
k kk k k
11
1
12 2 2
...
( )k
kk 12
...
1 )(
1
k kp, p(k) = product of the first k primes
... 11
81
k kk ( )
... ( )( )
log
11
2
11
1
21
1 1
kk
k k
k
k k k
...
0
2 )!2(!
)1(2)2(2
k
k
kkJ
...
1
202
4 )()2(
36 k k
k Titchmarsh 1.2.2
= 17
12
.7060568368957990482... 3
21 1
1
2 120
log( )x
dx
... )6(60
137
... 42
)1)(2
kk
k
.70673879819458419449...
2
1)()1(k
kk
.7071067811865475244... 2
2
1
2 4 12 12 cos cos sin
= ( )
11
4
2 1
8
2
0 0
kk
kk
k
k
k
k
k
=( )!!
( )!!
2 1
2 21
k k
k kk
=
01 36
)1(1
36
)1(1
k
k
k
k
kk J1029
= 11
2 11
( )k
k k
1 .707291400824076944916...
1
31
3
20
3 )()(
)6(
)3(
kk k
klc
k
k
Titchmarsh 1.2.9
... 1
21 2
11
2
log ( )k
kk
k
15 .707963267948966192313... 5
.70796428934331696271...
kLi
kk
k
kk
111)1(2
212
... 7
21
7
11
2 1
70
k
kk
k
k
( )
... 2/1)2(
1
1212
)!2(
2)1(
2
1cos11sin H
kk
kk
GR 1.412.1
6 .70820393249936908923... 45 3 5
...
0 13
)1(
3
2,
2
1
6
1,
2
1
6
1
k
k
k
.70849832150917037332... 2
12sin
)1(
2
1sin1cos22log
2
1(cos
2
1
1
1
k
kk
k
... dxx
x
1
0
2/11
11024
231 GR 3.226.2
.70871400293733645959...
2
11
kkk
... 2log2
92log
2
)8(324
82
2
G
=
124k
k
kk
H
... e
1 .70953870969937540242... 2 3
20 1
120
e k
kk
, ,( )!
.70978106809019450153... 8 2
1 1 21
1 2 1 22
40
( ) ( )cos( ) ( )
i e i ex
xdxi i
... 3/15
...
12)1(
2)2(24
k kk
= 1
0
2 )1log()log( dxxx
.7102153895707254988... sinh2
1201
164
3
kk
... dxx
x
0
2
cos12log4 GR 3.791.6
.71121190491339757872...
12/)3(2/)3(
122
2
1
2
1)1(
kkkkk
k Hall Thm. 4.1.3
3 .7114719664288619546...
31 1)(
kk kF
... )3()2(22
)5(9)2,1,2(
MHS
... coth
42
3
... 2
13tan
133
4
... 1)()()(2
0
kkkk
...
22
)(
1 1 22
)(2
1
221
)4(
)2(
k
k
k s ks
k
kkk
k
...
1 3
)1(14log3log1
kk
k
k
...
22
1log
1
11)()1(
kkk
k
k
k
k
k
kkH
... 3
24
1 1
12
12
1
( )!( )!
( )!( )!
k k
k kk
...
0
3
3
0
2sin2
)1(2
2
3
x
x
k
kk
k
k
... 4log7
...
03 12
)1(
k
k
k
...
0
22
)!2(1
211
2coth
2 k
kk
k
B
e
J948
=
0 1
2/sindx
e
xx
...
12 14
21
2coth
2 k k
J948
...
3
2
3
1
2
3log33log3
2
ll Berndt 8.17.8
... cos ( ) sin ( ) sin1 1 1 1 1si ci
= log cos( )x x dx10
1
.71317412781265985501...
( / )
( / ) ( / )
( )1 2
5 6 1 31
1
31
k
k k J1028
... 2
2
0
1
16
2
2
1
4
logarctanx xdx
1 .71359871118296149878...
1
)13(k
k
= 7
5
... G2
...
12
2
2kk
k
k
H
...
1
02 )21(
arcsin
3
31log
2dx
xx
x
... 1
0
1
02
sinh1
cosh
4
1
4
3dx
e
xdx
e
x
e xx
...
222
11
2sin
22
k k
...
0
3/1
3!
)1(
kk
k
ke
.716586630228190877618... dxx )41log(22arctan5log1
0
2
... 27
... dxee
eexx
x
1
0
2
2
1log
2
1
...
22
1)(
kkk
k
8 .7177978870813471045... 76 2 19
... 33
32 CFG G1
.7182818284590452354... ek k k k kkk
21
3
1
1 200 !( ) !( )( )
=1
1 20 ( )! !k kk
1 .7182818284590452354...
0 1
2
)2(!)!1(1
k k kk
k
k
ke
2 .7182818284590452354...
0
/2
!
1
k
i
kie
= dxex
x
02)/1(
log
3 .7182818284590452354...
0
3
)!1(1
k k
ke
...
2
12
1log2
csch2
logii
=
1 12
1
2
11log
2
)2()1(
k kk
k
kk
k
... 22
1
3
1
2
1
4
3log
2
3log2log
12 22
22
LiLi
=
0 2xe
x
... dxe
xx
02
3
)1()4()3(6
...
13
1 133
)13(
kkk k
kk
.71967099502103768149...
2 !
)(
k k
k
... 11
48 12 50
log
( )
x
xdx
...
12 244
1
2tanh
4 k kk
J950, GR 1.422.2
= dxx
x
0 2sinh
2sin GR 3.921.1
=
0
sinxx ee
x
... H
kk
kk
( )3
1 2
... )2(2)3(5
...
0 )3/1)(1(
)1(
2
3
k
k
kk
.72072915625988586463...
02 5
1
2
19tanh
19 k kk
... 4
arcsinh
... 5
)3(3
... e4log4log
1 J153
8 .72146291064405601592... 22
k
k
k( )
...
1
1
23
)1(3
2arccsch
7
3
7
4
7
3
k
kk
k
k
... log
=13
181
1
1
2
40
log( )( )
xx
xdx
... )!1(
1
2
3
2
2
1
2/3
k
e
k
k
... 21 3
16
2
8
1
1
2
20
1 ( ) log log( ) log( )
x x
xdx
1 .72257092668332334308... 18
3
...
222 )13(
3
3
1)(
2
333log9
kkk kk
k
... dxx
x
0 sin2
sin
33
4
.72330131634698230862...
22 )(
1)(
k k
k
...
1
3
)!1(
3
3
12
k
k
k
ke
...
13
12 1
)1(4
)3(3
12 k
k
k
k
.72475312199827158952... ( ) log sin3 2
8 30
x x
xdx
1 .724756270009501831744...
1
K
4 .7247659703314011696... 105
16 3 , volume of the unit sphere in R7
.724778459007076331818...
1
1
1
1
2!
2)1(
!)!12(
)1(
2
1
2 k
kk
k
k
k
kk
kerfi
e
.72482218262595674557... 1
2
2
3 3
2
3
1
3 112
1
cot( )k
kkk k
1 .72500956916792667392... dxxx
x
0
22
)1)(2(
log
6
2log GR 4.237.3
.72520648300641149763... 150
97
40
, probability that three points in the unit square form an
obtuse triangle
.725613396022265329004...
11
)1log(
kkk
e
H
e
eee
... 2/
0
)tan3log(2
3log dxx GR 4.217.3
...
2
1
2
2
0
tanh sinsinh
x dx
x
.726032415032057488141... 3
1
8
... log log2 23 2
...
0 2
2
2cosh
2sinh
)1log(4
2 dxx
x
x
GR 4.373.5
18 .726879886895150767705...
2
3 )1()()3(6)2(7k
kkk
=
232
23
)1(
1458
k kk
kkk
.72714605086327924743...
1
22
22
2
2sin
2 k
ii
k
keLieLi
i
... )4()2(
= 11
8
... 1
2
2
42
0
2
sin
cos (cos ) sin/
x xdx
...
0
/1
!
)1(
kk
k
ke
...
5
32Li
.72760303948609931052...
2
312
7
23
1 2
( )( / )
k
kk
1 .72763245694004473929...
1 2!
11
kkk
.72784877051247490076...
2 !!
11
k kk
... sinh
1
... 2
.728102913225581885497... dxxx
x
124
2
14
3log
34
...
... log ( )212
21
2
3
23
2
3
Li
= log
( ) ( )
2
30
1
1 2
x
x xdx
1 .728647238998183618135... 2)( xofroot
... 7
30 1
4 4
21
log
( )
x
xdx
.7289104820416069624... 21
64 2 14 30
dx
x( )
... cosh
1
.72908982783518197945... 45 5
64
4
31
( ) log
x
x xdx
... 221 e
... e )2sinh2sinh(cosh)2sinh2cosh(cosh2
1
=
0 !
sinh
2
1 2
k
ke
k
keee
1 .7299512698867919550...
2 1
2
cosh(log ) cos( log )i
.73018105837655977384... 22
12
0 )12(2
1
2
22
28
2log
kk
kGk
k
k
Berndt 9.32.4
= 4/
0
cot
dxxx
= 4/
0 2cos
tan4/ dx
x
xx GR 3.797.3
= 4/
0
)sinlog(cos
dxxx GR 4.225.1
= 4/
0
)cottanlog(
dxxx GR 4.228.7
=
1
02 )1(
arctandx
xx
x GR 4.531.7
1 .730234433703700193420... dxe
eerf
ex
x
0
4/1
2
2
11
2
.730499243103159179079...
3/2
3/1
.731058578630004879251...
0
)1(2
2/1tanh1
1 k
kk ee
e J944
... 2
27 130
log x
xdx
= dxee
exxx
x
03 )1(
)1( GR 3.411.26
...
2
)1)()1(k
k kk
1 .73137330972753180577...
2 21
1
kk
1 .73164699470459858182... 3
...
1 2!2log
2
1
kk
k
k
HEie
.731796870029137225611...
2233
1 )1(
3)1)3(3
kk kkkk
2 .731901246276940125002...
1
7
/1
7 )!7(
)(
k
k
k k
e
k
k
.732050807568877293527... 3 11
2 1
2
0
( )
( )
k
kk k
k
k
1 .732050807568877293527...
0 0
2
6
2
6
1
3tan3
k kkk k
kk
k
k
... 12
5tan32
AS 4.3.36
21 .732261539348840427114...
04
442
)1(
log
45
7
3
2dx
x
x
... 1
... 1
...
2
1
2
12log
2
1
8
2log3 22
iLii
iiLiG
=
12
2
11
1log
x
dx
x
x GR 4.298.14
...
22 1
)(
k k
k
.733333333333333333333 = 11
15
=
1 1 )3(4
12
)3()!2(
!)!12(
k kk kk
k
kk
k
...
2232
1 )1(
11)13(
kk kkkk
...
1
,1
11csc
...
1 0 )(2
1
kk k
... 16 16 22
32 3
2 12
3 121
log ( )( )!
( )!
k
k kk
... 2 12
12
1
( )!
!
k
k kk
... e
x xdxecoslog coslog
1
2 1
.73462758513149342362... 2
2 21
1
42 41
sin sinh
kk
... 3
4
CFG G1
...
1
3
)5(!11033000
k kk
ke
... ( )k
kk 21
...
1
22
12
/1
)1(1csch1
2 k
k
k
.73575888234288464319...
1 1
5123
!
)1(
)!1()1()1,2(
2
k k
kk
k
k
k
kk
e
3 .7360043360892608938... 2 11
21 4
40
/ log
xdx
...
1
2 )1)1(k
k kH
.73639985871871507791...
11 21
4!)!12(
!)!2(
3
1
27
32
kkk
k
kk
k CFG F17
=cos
( cos )
/ x
xdx
2 20
2
... ( )( ) ( )
11 1
2
k
k
k k
k
...
1 22
/1 2
)!1(
1)2(
k k
k
k
e
k
k
= 19
14
... )3(3 ... ( ) ( )( ) ( )2 2i i
... ( ) /k
kk
kk k
1
2 3
1
2
3 2
2
arctanh
...
0
22
4/sinh
4/cosh)1log(12log
281624
x
xx
GR 4.373.6
1 .7380168094946939249...
1 1
22
3
)(
13k kkk
kk
...
3
23Li
... p
kk
k !
1
...
2 6
)(5)1(
3
2log5
4
3log5
34
51
kk
kk k
.73879844144149771531...
1
1
2
)1()1(
2
3log
3
1
kk
k
k
k
.7388167388167388167 =
2
1,6
693
512
.739085133215160641655... xxofroot arccos
.7391337000907798849...
13
1
33
2 sin)1()()(
212
1
k
kii
k
keLieLi
i Davis 3.34
19 .739208802178717237669... 22
... 1
1
1
1
1
12k kkk
j
j kjk
( )
.740267076581850782580...
12 3
1
6
13coth
6
3
k k
J124
7 .740444313946792661639...
1
4
21010 !!)2()2(3)2()2(2)2(2
k kk
kIIIII
... 23
CFG D10
.740579177528952263276...
3
21
2
2
)()1(
24
52log
4
)3(7
kk
k kk
=
132
2
)12(2
21118
k kk
kk
=
1
2
427
20
kk
k
1 .740802061377891359869... 31log3
...
3
1
3
)1(
3
1
6
3
2
3log3
112
hgk
kk kk
k
= dxx
x
1
02
3)1log(
... dxe
xx
1
0 1
sin
.741227065224583887196...
122
2
)2/1(
1
2coth2
32
k kk
...
22
3
)1(24
2log5
8
7
kk k
k
... 14
...
2
1)( 1)1(k
kk e
...
2
2 2)1()(k
kkk
... 22
2 10
1
Gx
x xdx
log arcsin
( )
=arccosx
xdx
10
1
GR 4.521.2
=x x
xdx
sin
cos
/
10
2
GR 3.791.12
=log( )1
1 20
1
x
xdx GR 4.292.1
... pf k
ek
( )
0
...
1
21
)()2(log
)()2(
kk k
kk
k
k Titchmarsh 1.6.2
.744033888759488360480...
1 )12(2kkk
k
2 .744033888759488360480...
1
1
11 2
)(
222
1
12 kk
kkk
kk
kk
=
1 )12(2kkk
k
... 2
)3(G
...
1
3
3 5/sin
125
3
k k
k GR 1.443.5
.74430307976049287481... cos/
x dx0
4
.74432715277120323111...
1
1
12)1(
2
1arcsinh
55
8
5
2
k
k
k
k
...
log
5 .74456264653802865985... 33
... = 2
204 2 2 1
log xdx
x
... e
e
8
1
8
5)1sinh21cosh3(
4
1
=
1
2
)!12(k k
k
... )(log)(2
kkk
... 5
3
1
5
2
0
( )k
kk
k
k
... 1sinsin
... 1)2()(1
kkk
7 .7459666924148337704... 60 2 15
... e8
...
1
6
/1
6 )!6(
)(
k
k
k k
e
k
k
9 .7467943448089639068... 95
...
21
1
2 10
erfk k
k
k
( )
!( ) J973
= e dxx2
0
1
... 2
32 2 4
1
2
22
2
2
12
20
1
loglog
( )Li
x
xdx
.747119662029117550247...
( ) ( )
( )
k k
kk
1
2
2 .747238274932304333057... 1
22 5 15 6 5
= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22
... 59535 7 62
24 1
6 7
30
1 ( ) log
( )
x x
x
... G3
...
1
2
!
2)2(2log
k
kk
k
HEie
... )2()2(2
1coth
22 ii
= )1()1(2
1iii
i
= 2)12()2()1(1
1
1
23
kkkk
k
k
k
k
...
2 1
128
1 3 1 1
4G
kk k
kk
( ) ( )( )
... 8
7
...
1
41
4
1
3
1)4(
x
dx
kk
.74912714295902753882... 3 3
3
1 21 12
1
( ) ( )!
!
k k
k
k
k
... 2/e
...
1 )1(2
1
kk k
.750000000000000000000 =
1 3
)2log(sin4
3
kk
kii
=
22
1 1
11)2(
kk kk J369, J605
=
12 2
1
k kk K133
=
13
2)22()2(
2
1
kk
kkkkk
= 11
3 21
( )kk
=
13
4log
x
dxx
1 .750000000000000000000 = 7
4 21
H
k kk
k ( )
=11
4
1
1
2 4 6
2 6 8
p p p p
p p pp prime
8
372 .750000000000000000000 =
1
6
3kk
k
.75000426807018511438... 4
33
1
21
20
1
arctan tancos
cos
d
...
1 )()()1(
k ekk
eehge
... 2
22
1
20sintan
kk
k
...
2
3
4)( log
)3(k
iv
k
k
... 4/1
... 5 3
8 2
2
1
( ) ( )
H
kk
kk
=
log( ) log( )1 1
0
1 x x
xdx
... dtt
kk
k
0
2
13
13
4tanlog
)2/1(
)1(
4
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
...
1
22
1
22 1
2
121
log ( ) ( )! ( )e
Eik k k
kk
k
... HypPFQk k
k
k
{}, , , ,( )
( )!( )!
1
2
1
21
1
16
1
2 20
...
14
2/1
0 2!
)4(
k
k
kk k
e
k
k
k
k
1
2
2
2
3
1
27
34
36
1 )2(3
=
2
31
2
31
3
1
2
)3(33
iLi
iLi
2
31
2
31
3
333
iLi
iLi
i
= 1
06
2
1
log)1(
x
xx GR 4.261.8
... 23
96 12 40
log
( )
x
xdx
... )4(34
... )!1(
2
)!1(
122)2(
001
2
k
k
k
k
kk
kIe
kk
122
11
k kk
... e Ik
k kk0
1
1
21
2 1
2
( )
( )! !
!!
... e I e dxx0
0
11
2
2
cos
...
22
1
21 1 2 1 2coth i i
= 2 1
21 2 2 2 13
1
1
1
( )( ) ( ) ( )
k
k kk k
k
k
k
k
... 15
25 2 3
1
3
21
( ) ( ) ( )( )
H
kk
k
(conj.) 18 MI 4, p. 15
.754637885420670280961...
e
xe
dxxee
e
0
11
... 6
2
31
126
0
1
loglogx
xdx
... 3log3 2
...
2 21
1 120
log( )
( )(
k
)k k k
...
2
15
2
15log
10 22
2
Li
Berndt Ch. 9
1 .75571200920378862263...
1
/1
1
1!
)5( 5
k
k
k
ek
k
... sinh2
2 412
14
kk
...
0 )!2(k
k
k
B
...
0 )1(2)2log(2
x
dxEi
x
.75735931288071485359... 235
.7574464462934689241... 2 5 2 3 34
2 1
2
41
( ) ( ) ( ) ( )
( )
( )
k H
kk
k
... 1
120 e
k
k
... 9
4)3(
.757672098556372684654...
1
22
)12()2(8
1
2
)3(
8 k
kkk
=
223
2
)1()1(
1
k kk
k
... )()22()()22(4
1
4
)3(333 iLiiiLii
= dxxx
x 21
04
log1
1
...
0
)1(
1 kk
k
... 11 2 2 1
0
1
eEi Ei e e x dxx( ) ( ) log log( )
...
3
2
2
3log3
6
3
2
3hg
... 2 3 2 ( ) ( )
...
1
2 1)2(1)2()3(2k
kk
... x dx
e ex x
1
...
... 4/13
... 2/2
2/2
2
2
1
16
1
12ii eLieLi
= 2
cos)1(
12
1 k
kk
k
.75997605244503147746...
1 12 )!(
1
!
)(2
k jj
k kk
ke
.7602445970756301513... cos( )
( )!
1
2
1
2 20
k
kk k
AS 4.3.66GR 1.411.3
.7603327958712324201...
1 5
11
kk
...
1 )!2(
),2(1
k k
kkS
.760345996300946347531...
1 16
)1(
16
)1(1)32log(
3
1
k
kk
kk J83
=
1
1
1 22
)1(!)!12(2
!)!2()1(
k
kk
kk
k
kk
kk
k J267
=
0 !)!12(
!)1(
k
k
k
k J86
=
22 3x
dx
1 .76040595997292398625... 16 232 2 1
16 1
2
4 30
log( )!
! ( )
G k
k kkk
...
1
2
5
10 )!()2(4)2(5
k k
kII
...
2
1 1)1()()1(k
kk
.76122287570839003256... cot2
...
13
1log
1
k k
k
k
...
4
335
4
33
4
35
4
33
32
11
iiiii
=
02 1
)1(
k
k
kk
... u k
kk
( )
!
1
...
2
/1
1
1
!
1)1(
k
k
k k
e
k
k
.761390022682049161058...
2 !
)(
k k
ke
...
0
2sinh)1(4
2
dxxee x
... )1(log
... iie
etan
1
11tanh
2
2
J148
=
1
2)1(21k
kk e
=
1
2
)!2(
)14(4
k
kkk
k
B AS 4.5.64
= 1
02cosh x
dx
.761747999761543071113...
03
3/1
233
2
x
dx
... 2
3sin
...
1
1
!!
)1(
k
k
k
...
( ) ( ) ( )( / )
1 33 2
41
1
3
1
54
1
3
4
9 3
26
33
1 3
k
kk
2 .76207190622892413594... 2 3
30
12
3
2
6
1
2
1 2log log log( ) log
Li
x x
xdx
... cosh( )
22
4
2
2 2
0
!
e e
k
k
k
AS 4.5.3, GR 1.411.2
=
0
22 )12(
161
k k J1079
2 .762557614159885046570...
1
71
2!)1(
128
583
kk
k
k
k
e
...
2
0
2
)()(
kk
kk
1 .762747174039086050465... dxx
x
e
dxx
0 1
2
arcsinh
11arcsinh221log2
...
44
)3(
4
)()3)(2)(1(
4
3
kk
kkkk
...
1
3
2)!1(4
32
kkk
ke
... 2 1 1 11
2 10
sin cos( )
( )!( )
k
k k k
...
4
3
4
12log72log6 2 ll Berndt 8.17.8
... 53
.76410186989382872062...
12
2
)4/3(
1
9
168
k kG
...
2
)(1)(k
kk
.764403182318295356985...
1
42
42
4
4 sin
4
1
180 k
sii
k
keLieLi
.764499780348444209191... 1
2 1
1
2 11
4 1
2 4 11
1
1
12k
k
k
kk
kk
k k
( )
( )( )
( ) Berndt 4.6
=
1 )12(2
11
kkk
...
1
2
20
24 )(
)4(
)2(
72
5
k k
k
Titchmarsh 1.2.10
1 .764667736296682530356... )4()3()2(
= 17
13
.7649865348150167057...
sin sinh/ /
3 3
2 31
31 4 1 4
2 42 k
...
0 02200 )!)!2((
)1(
4)!(
)1()()1(
k k
k
k
k
kkiIJ
...
1
3
3 3/2sin
81
2
k k
k GR 1.443.5
... 11
1 21
k k kk ( )( )
... 1)()1( 5 kkk
... 3
2
5
3 20
log/
dx
ex 3
...
1
02)1(
)2()1(1
21 dx
x
eEiEi
e
e x
... 11 13 2
3 31
coth( ) ( )
csch ik i k ik
=
0
2
1
sindx
e
xxx
...
1 )!1(1)1(12
k
k
k
kHEi
...
1
2)2(logk
k
1 .767766952966368811002... k
kkk
k
k
k
k
kk
8
2
2!)!2(
!)!12(
22
5 2
01
2
1 .76804762350015971349... 3/23
3/13
3
)1()1(3
2
381
8 LiLi
i
=
03
21
02
2
1
log
1
log
x
dxx
xx
dxx GR 4.261.3
5 .7681401910013092414...
12 1
21
k k
.76816262143574939203... 61 3
86 6
21 3
23
3 3Gi
Li i Li i
( ) ( )
( )
=arcsin3
20
1 x
xdx
... 13
82
4
2
22
( ) coth( ) k
kk
= ( )2 11
16 2
2
kk kk k
... 2 22 12 1
1
/ ( ) /
k k
k
...
0
33/2
cos3
1
26
13 3
dxxe x
... =
112
11
kk
... 3G
.7687004249195908632... =1
20 ( )!! !!k kk
... cos ( ) cosh ( )( )!
/ /2 1 2 12
41 4 1 4
6
0
k
k k
2 .768916786048680717672... arcsinh12
1
2
20
log( )x
xdx
...
2
0 cos24
cos
3
21 dx
x
x
... i2Re2logcos
... 2 2 4 6 2
0
1
log arccos logG x x dx
31 .769476621622465729670...
1
4
32
3log15
61
411
kk
kHk
.769837072045672480216... 22
222 csch4
2sinhcsch8
cschcoth4
1
=
1
1
2
1)2()1(4
1)2()()1(
k
k
k
k kkkkk
=
122
2
)1(k k
k
... 8
7)2(
.76998662174450356193... 2 2 11
1 211
0
Jk k
k
kk
( )!( )!
.770151152934069858700...
0
2
)!2(
)1(
2
1
2
1
2
1cos
k
k
k J777
.770747041268399142066...
0 2!)1(2
1
kk
k
k
B
e
... 4
11
4
2
22
2
14 1 2
2
20
eK
x dx
x/ ( )sin
= GR 3.716.9 2/
0
2 )tan2(sin
dxx
.771079989624934599824...
1
2
4
)2(
kk
k
... 3
231
k
k k
kk
1 .771508654754528604291... )3(4)2(4
.771540317407621889239... 2
1cosh
1 .7720172747445211300...
12 1k k
k
.772029054982133162950...
sinh2
1
.77213800480906783028...
2 2
1
2
1
2 11 2
40
ex
xdx
/ cos sincos
1 .772147608481020664137... 2log12
)3(2
...
1
2
=( )
( )
k
kk
!
!
1
2
1 1
... 4 2 22 1
2 1 21
log( )!!
( )! ( )
k k
k kk
=
22
0 2
1)(
)2(2
1
kk
kk
k
k
=1
1120 ( )(k kk )
= dx
e ex x
1 20 /
= 1
0
1
0 1dydx
xy
yx
...
1
1
212log4
kk
kH
... )1(4
112
2
)1(2log4
01
kk
kHkk
kkk
k
=
0 )2/1)(1(
1
k kk
=
222
)(
kk
k
=
03
4 2sindx
x
x
... dxx
x
1
0
2/9
1256
63 GR 3.226.2
... 1 1
12 p p pkp primep primek
( )
= )(log)(
2 1
skk
k
s k
1 .773241214308580771497...
1
2
)!2(
22sinh22cos2
4
1
k
k
k
k
... 7
3)3(
... 4log6
...
1 1
1
k kk FF
... )sin( e
...
0
4
!15
k k
ke GR 0.249
... 3
264
1
1 4
1
1 4
3 1
3
1
31
( )
( / )
( )
( / )
k k
k k k
... 1
2 0
sinh sin sinh
x x dx
... 3
5
1
6
2
0
( )k
kk
k
k
1 .774603255583817852727...
1 82
2224464
kk
kk
...
0 2
)1(
kk
k
k
...
2
1
12
5
6
3 1 1
62
( ) ( )( )( )
k
kk
kk
8 .7749643873921220604... 77
...
1
1
0 !!
)1()2(1
k
k
kkJ
...
21
1
kkk
k
3 .776373136163078927203... )3(
... dx
x
x
20
1
2 .77749955322549135074...
215
/1
)!5(
)(
kk
k
k
k
k
e
... 2
1)1(
1)1log(1
6 22
2
eLi
eLie
=
02 )!1(
11
k
k
k
B
eLi
=
1
0 1xe
dxx
= 9
7
17 .77777777777777777777 = mil/degree
14 .778112197861300454461...
0
12
!
22
k
k
ke
... 4/110
1 .77844719779253552618... pf k
e kk
( )
!
0
... tan ( ) ( )
( )!
1
2
1 2 2 1
2
1 2 1 22
1
k k k
k
k
B
k
.778758579552758415042...
11
1 1cos1
)!2(
)2()1(
kk
k
kk
k
.778800783071404868245...
0
4/1
4!
)1(
kk
k
ke
1 .778829983276235585707...
02
2
2 1
coslog
1
2log
x
x
e GR 4.322.3
1 .779272568061127146006... )3(
...
3 3
2
3
2
3 10
1
arctan
e
e
dx
e ex x
...
1 )3(
11
k kk
... 3
tanh
.7808004195655149546... k
ekk
11
...
1 2
)(
kk k
k
...
0 !k
k
ke
=nn
n
loglog
)(lim 1 HW Thm. 323
.781212821300288716547... )1(1Y
.781302412896486296867...
122
)1()1(2 )13(
1
)23(
1
3
2
3
1
9
1
k kkg
J310
= dxxx
x
1
021
log GR 4.233.1
.78149414807558060039... 2
221
k
k
k
.781765584213411527598... 2
3log3
3210
21
3
8
6 .781793170552500031289... )2()(13
)3( 22
kkk
=
223
23
)1(
124
k kk
kkk
1 .78179743628067860948... 2 11
6 55 6
1
1
/ ( )
k
k k
.781838631839335317274...
1
2/1 12k
k
1538 .782144009188396022791...
4
1)3(
113 .782299427444799775042...
1
2/2
!
sinh2
2
1
k
kee
k
kee
6 .78232998312526813906... 46
1 .7824255962226047521... 1
0
2log]1},2,2,2{},1,1,1[{2
xe
dxxHypPFQ
.78245210647762896657426... F kkk
( )2 12
... 1
6
8 2
9
33
1
log log( )x
xdx
...
1
1)1(
kk
k
k
...
2
1)(
k kF
k
=
1
41
2)1(
125
598
kk
kk kF
2 .784163998415853922642... 2
2/3
.7853981633974483096...
0 12
)1(
2
1arcsin)1(
4 k
k
k
AS 23.2.21, K135
= )(Im1log(Imarctanh 1 iLiiii
=
1 )14)(14(
121
k kk GR 0.232.1
= ( )
( )sin( )
1
2 12 1
1
21
k
k kk J523
=
1 12
)12sin(
k k
k GR 1.442.1
=
1
2/sin
k k
k
= sin3
1
k
kk
GR 3.828.3
=( ) cos( ( ))
1 2
2 10
k
k
k
k
1 / 2 Berndt Ch. 4
= arctan( )
2
2 2 20 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.3
= arctan1
2 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
=
12 1
1arctan
k kk K Ex. 102, LY 6.8
= ( ) arctan
121
21
k
k k [Ramanujan] Berndt Ch. 2
=
1
2)12(
11
k k GR 0.261, J1059
=k k
kk
( )
( / )
1
1 2 21
J1061
=
02
02
02
022 14224)1( x
dx
xx
dx
x
dx
x
dx
=
041 x
dxx GR 2.145.4
=
03/24
023 21 x
dx
xxx
dx
= x dx
x1 40
1
=
log
( )
log
( )
x
xdx
x
xdx2 2
02 3
01 1
=
0xx ee
dx
=sin
sin
/ x
xdx
2 20
2
=sin4
20
x
xdx
=
0
sin1 dx
x
xe x
= dxxx2/
0
2 tanlog)(sin
= 1
0
logsinlog
)1(dxx
x
xx GR 4.429
= GR 4.533.1
0
)arccot1( dxxx
= GR 6/469/1 1
0
cossin)( dxxxx
1 .7853981633974483096... 4
12
2 30
k
k k
k
.78569495838710218128... cos sin( )
16 161
1
8 41
k
k k J1029
.7857142857142857142 =14
11
.78620041769482291712... 8
9
8
27
1
2 2 arcsinh
= 2 10
111
2
1
8
12
2
Fk
k
k
kk
, , ,( )
...
212 2
log
)!1(
1
2
log
kk
n
nkk
k
n
kk
...
1
4 1)2(k
k
... 1
11
112 1
1 ( ) ( )k k p pkk k
kp prime
... )2sinh2sinh(cosh)2sinh2cosh(cosh2
1e
=
0
/1
!
sinh
2
1 2
kk
e
ek
kee
.786927755143369926331...
1
51
5
1
4
1)5(
x
dx
kk
.78693868057473315279... 22 1
1 20
e k
k
kk
( )
( )!
... ( ) ( )log
( ) ( )2 2
1
2
3
23
1 23
2
20
1
Lix
x xdx
...
12/ 1
1
kke
Berndt 6.14.4
...
0
3
)!2(
3
9
4
k
k
k
e
... !5log
... csch
24
2log
2coth
42
1 2
2
12
12
1
2
1
4
1 iiii
=
)1(
1)1(
21
kk
kk
... 2/1)3()3(68 H
...
2 )(
1
k klfac , )...,,1()( kLCMklfac
...
1
3 1)2(k
k
... 15 5
25 4
2
( )( ) ( )
1 .78820673932582791427...
3 122
0 )(log)(log)(log)(n kkk
knkkk
... log log sin( )
( )!( )2 2
1 4
2 2
12
2
1
k
kk
k
B
k k [Ramanujan] Berndt Ch. 5
= 1
0
2
x x
xdx
cos [Ramanujan] Berndt Ch. 5
.788337023734290587067... coth
4
.7885284515797971427...
( )( ) ( )
( )
3
2
3
161 2 1 1
2
12
3
2 32
k k kk
kk k
...
22
22
log11log
1
1
1)(
kkk kk
k
kkk
k
=
1 21 2
111)(
n kn
n kn kk
Lik
k
... k
kk
k
1
12
( )
... 5
4
... 3log8
= 19
15
... =
1
1 !!
2222
k kk
kI
... ( )( )
log
11
2
k
k
k
k
...
1
2
43
4log44
81
188
kk
kHk
...
1 2
1
k kk H
.7907069804229852813... 1
1 2
1
21 0 1( ) !!
!!
( )! ( )!!k k
k
k kk k k
!!k
... H
kk
k
3
31 1( )
1 .791759469228055000813...
11 6
5
6
56log
kk
k
kLi
...
1
2
1
2/1
)1(1
2csc
2 k
k
k
1 .792363426894272110844... )1()1()(4
coth22
)1()1(22 iiiii
=
12 1k
k
k
H
.792481250360578090727...
1
4 222log2
3log3log
xx
dx
... 2
7
...
1 2
1
kk ke
...
2
2
.7939888543152131425...
( / )
( / ) ( / )
( )1 2
9 8 3 81
1
41
k
k k J1028
.794023616383282894684... e2 Berndt 8.17.17
.79423354275931886558...
0
)1()1()1(
1
sin)1()1(
2)(Im dx
e
xxii
ii
x
... kk
Hk 21
1)1(
... 1
42
20 0 2
1
I e Ie
k k
kk
( )sinh cosh
( !)
... )3(
3
...
0
3
)2)(1(2202log30
kk kk
k
.795371500563939690401... 5/1
.7953764815472385856... HypPFQ
kk
k
k
k
[{ , , },{ , }, ]( )
( )
1111
22
1
4
1
120
... 2
sin
.795749711559096019168... Lie
Lie2 3
1 1
.7958135686991373424... tan tanh1 1
4 .7958315233127195416... 23
...
1
2
3kkkH
.79659959929705313428...
Eik k
e
xdx
k
k
x
( )( )
!1
1 11
1 0
1
J289
= log x dx
ex0
1
= ( )
!( )
( )
( )! !
1
1
1
120
1
1
k
k
k
kk k k k
= HypPFQ[{ , },{ , }, ]1 1 2 2 1
8 .7965995992970531343... 8 1 12 11
4
1
Eik
k kk
k
( ) ( )( )
! Berndt 2.9.8
.796718392705881508606...
1
3 1)3(k
k
...
2
1)2(
)(
k k
k
... dxxx
x
0
222
)2)(1(
log2log
9
2log GR 4.261.4
.797267445945917809011...
112
)1(1
3
2log
2
11
kk
k
k
...
2
2
)1(
log
k kk
k
... 2
2
...
22
3)2()()1(
1)3(2
k
k
k
kkk
k
9 .7979589711327123928... 96
... 3 2 3
1
120
tanh
k kk
= 13
3
3 3
2
3 3
2
i i i
...
1 )!1(k
k
k
F
...
3
1 )(!
)1()1(
k
k kk
kk
.798632012366331278403... e
k k k
k
k
2
0
1
8
2
1 3
!( )( )
... )1(
1()1(2)2(1 2
=
2
2
2
1)()(
1
k
k
k
kkk
...
1
1
)1()!2(
4)1(2sin
2
2cos
2
3
k
kk
kk
.7989752939540047561... 2 1 1
1
2
120
sin ( )
( )!( )
k
k k k
... ( )
log2 1
2
11
1
21 12
k
k kkk k
k
...
2 12 22
)(
2
3
k jk
j
kk j
kkk
... 2
2log
sinh44
1
2
1
2
1
21
21
8
1 iiii
=
023 1
)1(
k
k
kkk
...
41 24
0
2
e xcos( tan )/
dx GR 3.716.10
.800000000000000000000 = 4
5
1
41
( )k
kk
J352
= F k
kk
2
1 4
.800182014766769431681...
1112
1
2
1sin
1
2)!12(
)2()1(
kkk
k
kkk
k
1 .8007550560052829915... log( )k
kk
12
1
.80137126877306285693... 2 3
3
( )
.801799890264644999725... 7 3
8
1
4
2 1
4
2
1
( ) ( )
k kk
k
12 .80182748008146961121... log9!
... ( )32
5
.80257251734960565445... 1
5
14
5 52
12
1
1
1
arccsch( )k
k k
k
... ( )
!
41
1
1
4k
kek
k
... 3 3
2 31
1
3 22
sin
kk
.8027064884013474243... si
k k
k k
k
( ) ( )
( )!(
2
2
1 4
2 1 2 10
)
1 .8030853547393914281... 3 3
22 3 2
1
1 30
( )( )
( )
( )
k
k k
=log2
0
1
1
x
xdx
GR 4.261.11
=x x
xdx
log
( )
2
20
1
1 GR 4.261.19
=log2
21
2
0 1
x
x xdx
x
edxx
= dydxxy
xy
1
0
1
0 1
)log(
.8033475762076458819... 1
2 211
kk
... iLi e Li ei i
2 22
22 / /
... ( ) /
1 1
1
1k
k
ke
... 6 3 3
... 3
212
3
... logRe log
5
22 i
... 11
21
kk k
... e
k
k
k
1
41
4/
... ( )
!
/3 1 1
1
1
1
3
k
k
e
kk
k
k
... 1
4 2
1
4 2
1
4
22
2 2
csch csc ( ( )h i
2 4 2
2 2
( )( ) ( ) ( )
ii i
4
i
=H
k kk
k ( )21 1
... 11
61
kk
... Lkk
k1 2
1
1
2
1
2/
.8061330507707634892... 4
9 3
121
kk
kk
= 4
3
5
3
2
1 3 2 31
k k
k kk
( )
( / )( / ) J1061
=
023 )1(x
dx
1 .8061330507707634892... 14
9 3
121
kk
kk
=dx
x( )3 20 1
... 2 2 2 2 3 3 2 2 3 3 2
1
32 2
2log log log log log log Li
= log log( )x x d 20
1
x
... dx
e x
x dx
e xx x
0 0
1 .80649706588366077562... k
kk !
10
... 2 2
2 21
1
2 2 21
sin
( )
kk
... 128 32 4 8 48 22 G log
=log( ) log
/
13 4
0
1
x x
xdx
... 21
21
/k
k k
...
22
2 )2(
k k
k
... 11
22
( )k
kk
2 .80735492205760410744... log2 7
... 4
3
1
3
1
31
3
0
1
, e dxx
... 1 3
2 .80777024202851936522... dx
x( )0
Fransén-Robinson constant
.80816039364547495593... ( )
( )
k
kk
1
2
2 .8082276126383771416... 4 3 2 ( )
= ( ) log1
1
2
0
1
x x
xdx
4 .8082276126383771416... 4 3 ( )
= log( ) log/1 1 2
0
1
x x
xdx
= dydxxy
yx
1
0
1
0
22
1
)log(
5 .8082276126383771416...
1
2 2)2()1(1)3(4k
kkk
.80901699437494742410... 2
1 5
4
.80907869621835775823... 2 21
21
log( )
k
kk
.8093149189514646786... 1
2
1
2,
... 1
3
3
2
1
3 1 11 3 2 3
cosh( ) ( )/ /
=1
5 3
2
5 3
2
i i
= 11 3
32 1
k
k
kk k
exp( ) 1
... H kk
kk
( )
1
21
.81019298730410784673... 1
0
42 log)1log())6()5()4((244120 dxxx
7 .8102496759066543941... 61
.81040139440963627981... 1
22
2
1
2 120
csch ( )k
k k
... 4
1 k
k
k
... e e i i / log2
.81056946913870217... 8 4
3 2
2 2
2 12 21
21
1
( )
( ) ( )
( )
k
k
k
kk k
... 36 2 31
1 6
1
1 62
1
2
1
21
( )
( / )
( )
( / )
k k
k k k
.8108622914243010423... sin
1
11
22 kk
.810930216216328764... 2 3 21
211
1
2 10
(log log ) , ,( )
( )
k
kk k
=dx
ex
1 20 /
... 15
15
2
1
420
tanh
k kk
... 2036
0
ek
kk
!
... log( / ( ))
( )
k k
kk
1
322
... arccosh
... 1
2
2
41
1 2
2 1
1
1
arcsinh( ) ( )!!
( )!!
k
k
k
k
=( )
1 42
1
1
k k
k k
k
.8116826944033783883... 4 2 2 2 12
22
2
log log
H
k kk
k
1 .8116826944033783883...
12
2
22log22log4
k
k
kk
H
1 .8117424252833536436...
1
22
1
4
)4(3)2,2(
jk jkMHS
.8117977862339265120...
H kk
kkk k
( ( ) )log
2 11
11
2
22
... 5
1
( )
!
k
kk
... 6
12
21
4
e
k
kk
k
( )!
... 2
3
e
... 1
2
1
2
1
2
1
22 2 2
2 2 2 ...
=256
3155
1
2
,
.81320407336421530767... 2
3
2
31
1
9 121
sin
kk
J1064
.81327840526189165652...
1
4
2 3
4
3
4 8
1 4
3 4
/
/ sin
1 .81337649239160349961...
... sinh
( )
2
2
4
2 10
!
k
k k GR 1.411.2
= 142 2
1
kk
GR 1.431.2
.81344319810303241352... 11
46 2 4 2
3
42 log log
( )
= ( ) log( )1 12
0
1
x x
xdx
... 4
9
768
18 42
1
log kH kk
k
1 .81379936423421785059... 3
1
3
2
3
hg hg
=d
2 20 cos
= log( )
log( )1
1
1
1
0
3
30
x xdx
x
xdx
=sin
(sin ) /
sin/ x
xE
xdx
1 4 220
2
GR 6.154
... ( )( )
12
k
k
k
k
.8149421467733263011...
2
313
2
3
2
sin k
kk
GR 1.443.5
... 3 2 2 3
3
1
1
4
30
logx
xdx
47 .81557574770022707230... 5
32 1
5 2
20
log x dx
x
=
xx ee
xiLiiLii
4
55
5
)()(2416
5
=x dx
x
4
0 cosh
GR 3.523.7
.81595464948157421514... 3
38
.8160006632992495351...
2
2
1
1
tanh(log )
3 .816262076667919561... 1
2
220 0
0
Ie
I ek
kk
cosh
( !)2
... 6
12
1
2 221
( )
( )f k
kk2 Titchmarsh 1.2.15
.81638547489578... ( ) ( ) ( )3 1 3 1 3 11
k k kk
.8164215090218931437...
( ) ( )
( )
2
2 1
2 1
2 1
1
2
1
212 1
12
12
0
k kk
kk
k kk k
k
... 2
3
1
8
2
0
( )k
kk
k
k J168
.81659478386385079894... 3 2
8
1
1 1
2 loglog
x
x
dx
x GR 4.298.12
1 .81704584026913895975... H
kk
k
( )
!
3
1
... 63
... 33 102
5
1
ek
kk ( )!
... ( )61
5
16
16
1
k
dx
xk
... 3
2
9
4 6
32
21
cos k
kk
GR 1.443.2
= 1
2 23
23Li e Li ei i
4 .81802909469872205712... e
=9
11
... arccsch1
3
... 2 2 14
2 11
1
sin ( )( )
k
k
k
k !
... 3
11
... log
coth
x
xdx2
0
GR 4.371.3
1 .81895849954020784402... ( ) ( )2 3
... 2 6 32 22 2
3
1
log( )
k
kkk
... 1
44 3
1
2 4
2
21
, ,log x
xdx
... 1
11 1e
k
ekk
kk
( ) Berndt 6.14.4
1 .82030105482706493402... ( )
( )!sinh
8 6
2 1
1
1
24
1
k
kk
kk k
... ( )21
kk
... 2 11
2
( )k
k
... x x
edxx
2
0 1
log
... ( )
!
/k
k
e
kk
k
k
3 1
1
1
31
... 2
12
=( )
1 1
21
k
k k J306
=H
kk
kk 21
= kk
k
( ) ( )
1
21
2
=x
edx
x
x xdx
x
xdxx1
1
02
1
2
0
1
log log( )
=x
edxx10
2
log
GR 3.411.5
= log( )log
11
0
1
0
x
xdx e dxx Andrews p.89, GR 4.291.1
=
log( )1 2
0
1 x
xdx GR 4.295.11
=logx
xdx
11
2
[Ramanujan] Berndt Ch. 9
=
log x
xdx
10
1
GR 4.231.1
=
log 10
1
e dxx
=log( )
( )
1
1
2
20
x
x xdx GR 4.295.18
= log1
1
2
2 20
x
x
x
xdx GR 4.295.16
= log 11
1
x
dx
x
=log( )
( )
x
x xdx
2
20
1
1
=x
edx
x
3
0
2
1
=e
edx
x
e x
2
0 1 GR 3.333.2
=xe
xdx
x
sin0
= dxx
x
02
2
cosh
... ( )
( )!sinh
4 2
2 1
12
11
k
k kkk
... ( )kk
k 2 12 11
.8235294117647058 = 14
17
.8236806608528793896...
22 2
062
x xd
ex
log x
1 .82378130556207988599... 18 3
22
( )
1 .82389862825293059856... 3
17
.82395921650108226855... 3 3
4
3
21
log sin
x
xk
.82413881935225377438... 1
2
2
0
2
cix
x( ) log
sin/
.82436063535006407342... e
k
k
k
k
kk kkk k2
1
2 2100 0
! ! (2 )!!
.82437753892628282491...
1
22
1)3()3(23
25
k
kk
= k
k kk
1
12 32 ( )
... 1
42 2 1
164 4
1
sin sinh
kk
... )1(4
3)1(
4
32log3
23
4
1
1
khgk
k
k
=
1 )34(
3
k kk
.8248015620896503745...
( )
!
2
1
k
kk
8 .8249778270762876239... 2
1 .82512195293745377856... J J J Jk
k
k
k3 2 1 0
5
20
2 6 2 7 2 21
( ) ( ) ( ) ( )( )
( !)
13 .82561975584874069795... ( )!
12 22
0
ek
k
k
LY 6.41
9 .82569657333515491309... 9
G
2 .82589021611626168568... k
kk !
12
.82606351573817904... ( )1
14 3 22
k
k k k k k
.82659663289696621390... k k k
kkk k
( )
( )
2 1
3
3
3 122 2
1
... sincos ( )
( )!2
1 4 1
1
21 2
2
1 2
2
k k
k k GR 1.412.1
...
2 3 2
2 242
2
2 log
log
= log sin2
0
2x xdx
x
GR 4.424.1
... 3 3
21
1
9 21
kk
GR 1.431
= cos
3 21
kk
= 0
1 3/
.8270569031595942854... ( )33
8
.82789710131633621783... 1
1
1
12
e ii /
... 3 5log
.828427124746190097603... 2 2 21
4 1
2
0
( )
( )
k
kk k
k
k
... 2 2 12 1
2 21
( )
( )!!
k
k kk
!!
... 1
3 320log
x dx
x
...
121
2 14
3
0
Lix dx
ex / 2
... log !
!
k
kk
2
.8287966442343199956... 14 3 161
22
12
12
32
( )( )
( )
kk
= x
e edxx x
2
20 1/
16 .8287966442343199956... 14 31
2
12
12
30
( )( )
( )
kk
... ( )k
kk 21
... 1
21 k kk !log !
... ( ) ( )
( )
( )3 2
1
1 21
k
kk
... i e
e
i i
i i
i
i
1
1
1 1
1
cos sin
(cos sin )
1
... log ( ) kk
2
5 .83095189484530047087... 34 ... H ( )
/2
2 3
... 1
2 2
2 1
122
1
e e
k
e ekkk k
cot( )
8 .8317608663278468548... 78
... 1
31
13
13
( )
( )
k
k pk p
... 2 10
Gx x
edxx
tanh
... 2G
=( !)
( )!( )
k
k k
k
k
2
20
4
2 2 1
Berndt 9
= x
xdx
sin
/
0
2
Adamchik (2)
=x
xdx
cosh0
=arcsin x
xdx2
0
1
1
=arctan x
x xdx
1 20
GR 4.531.11
= GR 6.141.1 K x dx( )0
1
= K x Adamchik (16) dx( )2
0
1
... 3
13
1
9
2
0
( )k
kk
k
k
3 .83222717267891019358... 1
49 3 3
1
1 10
log( )( /
k kk 3)
... 1
43 2
22
0
2
log loge xx dx GR 4.383.1
1 .83259571459404605577... 7
12
... 1 3/
... 45 5
81
0
1 3 ( )log( )
log x
x
xdx
...
log2
... 3
8 2 14 20
dx
x( )
= 5
4
7
4
1
1 4 3 41
k k
k kk
( )
( / )( / ) J1061
.83327188647738995744... Li2
2
3
.83333333333333333333 = 5
6
=( )
1
50
k
kk
=1
1 21 ( )( )(k k kk
3)
=1
4 2
2 2 1
2 21 1k
k kk
k
k
k
k k( )
( )!!
( )! ( )
.8333953224586338242... | ( )|( ) | ( )| k kk
kk
k
kk
1
2 1
2 2
4 11 1
... cos cosh ( )( )
1 1 14
40
!
kk
k k
1 .83377265168027139625... root of ( )x x
1 .83400113670662422217... ( )
( )!sinh
2
2 1
1 1
1 1
k
k kk k
k
... 15 90
90
2 1
4 1
2
4
( )
( )
... 1
3
12
3
2
1
30ee
k
k
k
cos( )
( )!
3 .83482070063335874733... pf k
kk
( )
!
0
... ( ) ( )sin
31
4 32 2
2
31
Li e Li ek
ki i
k
... 2
3 213
8 2 81
2
log/k kk
= 81
2
1
3
( ) ( )
k
kk
k
=( )
( )
k
k kk
1
2
2 121
!
!
... h k
kk
( )2
1
100 AMM 296 (Mar 1993)
.8352422189577681... 1
1 F kkk
... 31 28350 5
84 1
6 6
30
1
( ) log
( )
x x d
x
x
1 .835299163476728902487... 5
24 22 2 2 1 1
12
20
( log ) log( ) logG xx
dx
... 135
453 2 4
4
( )
...
3 3
2
3
1
6 5
1
6 21
log
k kk
J79, K135
=( )
1
3 10
k
k k K Ex. 113
= dx
x x
dx
x
dx
e ex x2 11
30
1
201
... 5
3 327
1
1 3
1
1 3
3 1
3
1
31
( )
( / )
( )
( / )
k k
k k k
1 .83593799333382666716... 1
1 ( !!)!!kk
... ( )k
kLi
k kk k2
22
1
1 1
... ( )
!
1
12
k
k k
.8366854409273997774... e ek e
ek
kk
1 1
0
1
1/ ( )
( )!
6 .8367983046245809349... 28
3 3
32 10
k
k k
k
... 4 1 0
... 4 21
22
0
csclog
/
x dx
x
... Cosh int1
... log( )( )
( )2 1
2
2 11
k
k kk
Wilton
Messenger Math. 52 (1922-1923) 90-93 ... log2
... cos( )
( )!
1
3
1
2 30
k
kk k
AS 4.3.66, LY 6.110
.83800747784594108582... 3
37
...
0
4
)!14()2()2(
1
2
sinh
k
k
kii
=
22
11
k k
... log2
1
2
2
e
e J157
=
1 !
2
k
kk
kk
B [Ramanujan] Berndt Ch. 5
.83861630300787491228...
1
3
)5(
)3(124
125
k k
k
... G2
... ek
k
k
k
( )( )
!
11
0
GR 1.212
... 5
11
1
1 6
1e
k
k
k
k
( )
( )!
... 531
e ,
... log arctan log3 2 21
21
22
0
1
x
dx
... 2
1 2
2 1 20
1
logarcsin
( )
x
x xdx GR 4.521.6
...
2 21
2
20
2
sin
sin
x
xdx
... 7 45
8
2
7
2
0
e
k k
k
k
!( )
.84062506302375727160... 2 2 1 11
14
40
logx
dx
.840695833076274062...
sin sinh/ /
2 2
21
21 4 1 4
2 42 k
... 23
5
( )
( )
... ( )
1
3 2
1
1
k
k kk
.84100146843744567255... 2
8
... 11
2
kk !
...
22
0
2
24
logcos ( )
sin
/
x x x x
xdx GR 3.789
2 .84109848849173775273...
1
0
1
0
1
0
4
1
log
240
7dzdydx
xyz
xyz
3 .84112255850390000063... H
kk
k
32
1
5 .84144846701247819072...
logcos
sin2 4
210
Gx x
xdx GR 3.791.4
= GR 4.4334.10 log sin10
x dx
... sin Im( )
( )1
1
2 10
ek
ik
k ! AS 4.3.65
= 0
1 /
= 2
1
211J ,
= 112 2
1
kk
= cos1
21k
k
GR 1.439.1
= 14
3
1
32
1
sin kk
GR 1.439.2
= coscoslog
x dxx
xdx
e
0
1
1
... ( )71
6
17
17
1
k
dx
xk
.84178721447693292514... 2 3 11
320
logx
dx
... 1
22 2
2
1 20
Eik k
k
k
( ) log!( )
= 16
19
... 1
4096
1
4
3
4 15 5
5
20
1
( ) ( ) log
( )
x
xdx
... 32
32
1
3
1
1 31 0 10
J Fk k
k
kk
; ;( )
!( )!
...
1 2
1
k kk F
... 1
3 3 331e
B
kk
kk
... erfk k
k
k
( )( )
!( )1
2 1
2 10
AS 7.1.5
= 2 2
0
1
e dxx
... 3 33 1
3 1
11
65
60
log( )
( )( )
k
k k k
... 3 3 4 23
212 42 2 log log log log
= l l
1
6
5
6 Brendt 8.17.8
.8432441918208866651... kk
k
2
1 4 1
... Gx dx
x 2 2
20
1
16
2
4
log arctan
=x dx
x
2
20
4
sin
/
GR 3.837.2
... cos ( )( )
( )!3
0
1 21
3
1
41
1 3
2
k
k
k
k GR 1.412.4
16 .84398368125898806741... e Ik
k kk
20
0
22 1
( )!
... 21
2
1 2
2
1 2
2 1 12
1
1 0
sin( )
( )!
( )
( )!(
k k
k
k k
kk k )k
... 483 2
2
261 2
2
917
4
2log log
=k H
k k kk
kk
4
1 2 1 2( )( )(
3)
... 120 6 360 5 39 4 180 3 31 2 1 ( ) ( ) ( ) ( ) ( )
= k kk
5
2
1 ( )
... Li3
3
4
... cosh( )!
2
4
0 2
k
k k
... Fk
k
k
k 21
=5481948
15625 4
4 2
1
k Fkk
k
... 1
4
1
40 1
1
4 10
, ,( )
!( )
k
k k k
... cos( )
( )!
1 1
20
k
kk k
AS 4.3.66, LY 6.110
... 5
7
1
10
2
0
( )k
kk
k
k
... 9 32
3
1
ek
kk ( )!
... ( )
( )!sinh
2
2 1 2
1 1
22 11 1
k
k kkk k
k
... 22
2
208
1
21 2
1 4
2 2 1
log( ) ( !)
( )!( )
k k
k
k
k k Berndt Ch. 9, 32
= x x
xdx
sin
sin
/
2 20
2
... 3 22
0
x x
edxx
log
=11
13
...
( )
( )
( )6
3 31
k
kk
... 7
621 3
2 1
70
k
kkk
... ( ) ( )
( )! /
1
2
1
22 1
1
k
kk
k
k
k k
e
.84657359027997265471... log log
cosh(log( ))2
2
1
212
23
1
x
xx
dx
x
... 2( ) e
... ( ) ( ) ( ) ( )2 3 2 22
k kk
=2 1
25 41
k
k k kk
3
... ( ) ( )2 3
... ( )
1
15 4 3 20
k
k k k k k k
... sin cos/
x x d0
2
x
... 1
21k
kk F
.8474800638725324646... ( ) ( ) ( )( )
1 1 111
!
e Ei eEiH
kk
k
=
( )
( )!
1
11
k
kk
=1
1
1
111
1
11m k
e
m km
k
k
k
m
k
k!( )!
( )
!( )!
AMM 101,7 p.682
.8474804859040022123... 1 2
31 ( !)k
k
kk
... 4 3
9
1
2 2
12
12
1
( )!( )!
( )!
k k
kk
2 .84778400685138213164... ( )
!
/k
k
e
kk
k
k
4
0
1
41
.84782787797694820792... 7
256
1
4
3
31
k
k
k
sin GR 1.443.5
= 1
4 41 21 4
34
34
i
Li e Li ei i( ) / / /
.84830241699385145616... 3 3 13
313
2
k
k k
kk
( )
.848311877703679...
iLi e Li e
k
ki i
k2
1
222
22
21
( ) ( ) sin/ /
... 8 2
3
32
1
log log( )
x
xdx
...
( )
( )??
( )
( )
k
k
k
k kk k21
12 2
= sum of inverse non-trivial powers of products of distinct primes
... 16
27 1
2
3 20
log
/
x dx
x
.8488263631567751241... 8
31
1
4 22
kk
9 .8488578017961047217... 97
...
eerfi
k
k k
k
k
k
kk
1 40 0
1
2
1
2 1
1
2 1 2/
( ) !
( )!
( )
( )!!
...
1
54 3
1
3
1
3 22
31
( ) ( )
( )
kk
... ( )k
kk 31
... 6
.849581509850335443955... ( )
( !!)
1 1
21
k
k k
...
14
0 )()(
k k
kk
... H kkk
2 11
2 1
( )
=1
2 11
11
11
12
22( )
log log logk
kk
kk kk
.8504060682786322487... 33 1
1 330
e k
k
kk
( )
( )!
... 8 2 12 2
0
1
G x
xdx
log
( )
... sec ( )( )!
1 12
2 k kE
k
... csch12
1
i
i e esin J132
= 21
2 10 e k
k
GR 1.232.3
= 12 2
2
2 12
1
( )
( )!
kk
k
B
k AS 4.5.65
.85149720152646256755...
He k eke
k1
1
11 1
1/ ( )
= ( )k
ekk
12
.85153355273320083373... e e ee k
k
kk
log( )
( )1
1
10
... i i i i
4
1
4
1
4
3
4
3
4
i
=sin
cosh
x
xdx
0
... 7 2log
... 22 1
2
1
22
1 2 1
1 21
log/ ( )
/
( )
( )
k
k kk
Berndt 8.17.8
... 1
43 3
1
2 3
2
21
, ,log x
xdx
... 3 2 4 3 1 32
( ) ( ) ( ) ( ) ( )
k
k
k k k
=2 2
1
4 3
4 22
k k k
k kk
( )
1
... 4
1
2
1
2
11
81 3 2 3 31
( ) ( )/ /
kk
... 4
3
4
32
1
3 21
log( /k kk )
= xe e dxx x
10
GR 3.451.1
.85369546139223027354... 2 24
31
2
2
12
log( ( ) )
k kk
k
=4 1
2 1 22
k
k kk
( )
1 .85386291731316255056... 1
21k
k k
... 5
2
... 1
4
1
42
1
4
1
2
5
41
12
2 1 40
Fdx
x, , ,
= K1
2
... 3 5 5
2
1
5 1
2
0
( )
( )
k
kk k
k
k
... dxx
x
02
161loglog12log24 GR 4.222
... ( )
( )!sinh/k
kk
kk k2 3
1
2
3 2
1
... 2 8
2
4 0
log log cosx x
edxx
... e e
ek
k
e e
k
1
1
121
/cosh sinh(sinh )
sinh
! GR 1.471.1
... bp k
kk k
k
( )
2
1
1 202
0
... 2
1
2
1
2 2 10
erfk k
k
kk
( )
! ( )
6 .85565460040104412494... 47
... ( )
!( )
11
3
2
k
k
k
kk
=k e k k
k e
k
kk
2 1 2
3 12
3 1/
/
... 2 2
52
0
ee x dx
sin x
...
11
11
0
ek k
k
kk
/ ( )
! ( ) )
... ( ) coth24
=6
7
1
60
( )k
kk
...
4G
... coscos1
... ( )k
kk 2 12
.857842886877026832381... 4 53
21
22
0
( ) ( )
x Li x dx
... 3
23 2
1
2 13
2
0
1
( )log
( )( )
Lix
x xdx
2
... 41
2
20
cos
( sin )
x
x
= dxx
x
1
0 1
arccos
= log( )cosh sinh
cosh1 2
20
2
xx x x
x
dx
x
GR 4.376.11
... 35
128 1
7 2
0
1
x dx
x
/
... 24 e
... 2 3
0
2
31
k
k
k
k
k
k
e
k
( )
!
... e Not known to be transcendental
... 2
1
2 1
2
( )
( )!
k
kk
... 96 354
2
1
ek
k kk !( )
... 1
1 231 kk
/
... 4
3
1
31 3
0
ek
k kk
/
!
.86081788192800807778... 4
5
1
2
4
5
1 5
2 21
arcsinh
logF
kk
kk
= ( )
( )
1
2 12 11
22
k
kk
k
k
dx
x x
.8610281005737279228...
2 2
21
4
1
221
coth
kk
J124
.861183557332564183... 1
42 2
1
2 221
( , , )log
x dx
x
.8612202133408014822... ( )81
7
18
18
1
k
dx
xk
.8612854633416616715... 3
36
...
1
0
1
0
1
0
1
0
2
14
3)3(32log8 dzdydxdw
wxyz
zyxw
.86152770679629637239...
11
2
1
20 1
1
2 1
1
1
, ,( )
!( )
k
k k k
.86236065559322214... 1
21k
k k ( )
20 .8625096400513696180... 1
21 2
0ee e
k k
ke e
k
/ cosh
!
.8630462173553427823... 34
3
1
3 1
1
1 30 0
log( )
( ) /
k
kk
xk e
.8632068016894392378... 1 1
21 k
k
kk
log
4 .8634168148322130293... 48 8
22
( )
3 .86370330515627314699... 2 1 312
csc
AS 4.3.46
.86390380998765775594...
1
3
)3(
)3(26
27
k k
k
.86393797973719314058... 11
40
6
60
sin x
xdx GR 3.827.15
.86403519948264378410... 1
83 4 2 42 4
4
1
e e ek
kk
cos coscossin cossinsin
!
... 6 9 12
9
4
9 93
1 31 3 1 3 1 3
// /sec sec sec /
[Ramanujan] Berndt Ch. 22
.8645026534612020404... 1
8
3
2
1
4
3
2
3
4
1
2 1 3 20
, ,( )
( ) /
k
k k
.8646647167633873081... 11 22
1
1
ek
k k
k
( )
!
2 .86478897565411604384... 9
.864988826349078353... 1
3 12k
k k( ( ) )
.86525597943226508721... e
1 .8653930155985462163... 7 2 12 4 18 3 1 ( ) ( ) ( )
= ( ) ( ) ( )
1 13
2
k
k
k k k
= 8 3
1
4 3 2
2 42
k k k k
k kk
( )
1
.86558911417184854991... ( )
( )!cosh
8 6
2
11
1
6
14
k
kk
kk k
.86562170856366484625... 1 2
.86576948323965862429... 22
1arctane g d
= arctan(sinh ) ( ) arctan( / )
1 11
1 21
k
k k
[Ramanujan] Berndt Ch. 2, Eq. 11.4
.86602540378443864676... = 3
2 3 sin cos
6
=( )
1
12
2
0
k
kk
k
k
... 1
2
5
10 5
1
5 120
csch
( )k
k k
... 1
4 22 1 2
1
4 10
log( )k
k k K135
=1
8 7
1
8 31 k kk
J82, K ex. 113
= 16
1
4 22 2 2 2 2 2
2 2
2 2
arctan arctan log
= 16
4 27
8
3
84 2
8
3
8
cot cot logsin logsin
=dx
x x
dx
x2 21
40
1
1
... 2
36 2
1
1 1 40
( log )( )( /
k kk )
... 21
2
1 22
1
21
arcsinh
( )k k
kk
k
k
5 .8673346208932640333... 1
128
1
4
3
463 3
3
0
( ) ( ) tanh
x x
edxx
11 .8673346208932640333... 1
128
1
4
3
43 3
3
0
( ) ( )
cosh
x dx
x
5 .86768692676152924896...
1
)(
k kF
k
...
1 2 ),2(
)1(
k
k
kkS
1 .86800216804463044688...
2 13 4 40
/ /
dx
x 2
.86815220783748476168...
13 3
108
1
432
1
6
5
62 2
( ) ( ) ( )
= 3 30
1
3 1
1
3 2
( )
( )
( )
( )
k k
k k k 3 J312
.86872356982626095207... ( )31
3
1 .86883374092715470879... )2,4(2268
)3(6
2 MHS
.86887665265855499815... ( )
( )log
1
2 11
1
21 1
k
kk
kkk
... ( )
( )!cosh
2
2
11
2 1
k
k kk k
... 1
22
22 2
1
e ek
ke e
k
k
/ cos
!
.869602181868503186151... 2
16
1
2
1
2
1
2
1
2
2 2
1 1
( ) ( )
4
2 2
16 2cot
= H
kk
k 4 221
... 2 2
0
1
1
logsin
sin
x
x
dx
x GR 4.324.1
... 21
1
2
1
3k
k
k
k
k
ke
( )
!/
.87005772672831550673... 4
2
2
2
0
log
loge xx dx GR 4.333
3 .87022215697339633082... I Ik
k
k
k
k
kk k0 1
3
21
3
0
2 22
2( ) ( )
( !) ( )!
.87041975136710319747... 21
32
1
2arcsin arctan
=( )
( )
( )!!
( )!!
1
2 2 1
2
2 1 30 1
k
kk
kkk
k
k k
...
4
23
1724 2 2 3
1
2 1
log ( )( )
( )
H k
k kk
k
... 1 4/
... 23 e
... 2
21
28 21
1
4 1
1
4 1
( )
( )
( )
( )
k
k
k
k k J327
= 1
64
9
8
7
8
5
8
3
8
=
log x dx
x40 1
... 5
16
1
16 2
3
1
e
e
k
kk
( )!
...
( )
( )
k
kk
1
12
... 29
207 1 ( )
... x dx
e ex x
2
1
... 15
... ( ) ( ) ( ) ( )
1 3 1 3 3 11
1
k
k
k k k 3
... 4 82)
3
1
ek
k kk
!(
... 4 11
4
1
ek
kk
( )!
... 4e
... 11
231
k
.87298334620741688518...
k
k
kkk
k 2
)1(6
)1(315
0
1 .8732028500772299332... 11 2
2 2
( )
( )
( )
( )
k
k
k
kk k
1
.87356852683023186835... 1
log J153
1 .87391454266861876559... 11
21
k
kk
... 3 2 2
20
2
20
2
8 2 2 1 2
log log logx
xdx
x
xdx
7 .8740078740118110197... 62
... 1
6 2
1
42 3 2
0
23 2
0
2
sin cos/
//
/
x dx xdx GR 3.621.2
... ( )
( )!cosh
k
k k kk k2 2
1 1
2 1
k
1
... ( ) ( )4 2 4 20
k kk
... 1
43
3
23
3 3
3 3 32
1
log log
H kk
k
.87446436840494486669...
( ) ( )
( )
int
k kk
nkk
a nontrivialeger power
nk
11
2 22
... 3G
= 7
8
1
72 1 2 1 1
0 1
( )( ) ( )
k
kk k
k k
... 3 2
082
x x
edxx
tanh
... 3 2
20
2
08 1
log
cosh
x
xdx
x
xdx GR 3.523.5
= 3
3 3
2
42
i Li i Li ix
e ex x( ) ( )
... cos( )
( )!
1
2
1
2 40
k
kk k
AS 4.3.66, LY 6.110
... G
... 2
21
2
1
21
log( )k
k k k
= log 1 2
0
1
xdx
x GR 4.295.2
= log( )
11
1 20
xdx
= log1
1
2
20
1
x
xdx
=x dx
x10
2
cos
/
= dydxyx
yx
1
0
1
0221
... coth
... 109
81 3
1 3 4
1
e k
k kk
/
!
5 .8782379806992663077... 3
82 4
1
2 2
3
21
, ,log x
xdx
... 2 21
24
1
24
1
2 2 10
sin cos( )
( )! ( )
k
kk k k
.879103174146665828812... ( )( )
log
12 1 1
111
12
1
k
k k
k
k k
k
... logk
kk2
1 1
...
1 )1(!
1
k kkk Related to Gram’s series
.87919980032218190636... 2
2 10
k
k kk
( )
.87985386217448944091... k
k kk
!
1
.8799519802963893219...
0
1 113 1
1
3 1
( )kk
kj k
jk
= 1
3111ijk
kji
.88010117148986703192... 2 11
1 410
Jk k
k
kk
( )
!( )!
1 .880770870194...
0
1 11
11
1( )
( )
k
k k k kk k
=1
111 jk jkkj ( )
... I0 3( )
... e
e e
k
kk
2 201
1 1
2
1
tanh ( ) J994
5 .88097711846511480802... 2 1
2
1
2 21
2
1
4 2F
k
kk
, , ,( !)
( )
2 !
... 0 1
2
0
41
26
1
3 2F
k k
k
kk
; ,( )
!( )!
... 9 3 12 2 3 31
1 5 60
log log( )( /
k kk )
= ( )k
kk 6 2
2
... cosh2
2
...
tanh/2
1
1 220
k kk
=
2
1
2
1 iii
... log(1 + 2) = arcsinh11
4 2 1
2
0
( )
( )
k
kk k
k
k J85
= log tan8
= arctanh1
2 i iarcsin
.88195553680044057157...
k
kk
12
21 2 2 1( )
1 .88203974588235075456... 1
121 k kk log( )
... 2
21 2
2 1
2 1
0
erfk k
k k
k
( )
!( )
=15
17
... e k
k
1
2
1/ !
...
1
3
3
63
1
322
csc
( )k
k k
... 2
4 11
k
kk
3 .88322207745093315469... 5 1 14
120
logx
dx
.88350690717842253363... ( ) ( )2 1 2 11
k kk
3 .8836640437859815948... e EiH
kk
k
1 111
( )( )!
... HypPFQk k
k
k
11 2 2 2 11 1
21
, , , , ,( )
( !)
... 4
81 3
3
3
g J310
1 .88410387938990024135... 64
10395
5volume of the unit sphere in R5
... e log 2
1 .8841765026477007091... 11
4
3
27
0
2
x e x dxx log
.88418748809595071501... H k k
kk
k
! !
( )!21
=
2
13
2
31
3
3log
3
2
33
222
iLii
iiLi
... 16
17
16
17 17
1
41
2 40
arcsinh ( )! !
( )!k
kk
k k
k
3 .88473079679243020332... 22
2
2
2
2
1
GH k
kk
k
k
log ( !)
( )!
... 2
31 3 3 3
3
2 10
cosh sinh( )
k
kk k
1 .88491174043658839554... Gx dx
x
2 2
2 20
1
32 1
log
( )
1 .8849555921538759431... 3
5
... 2 2
2 2212
1
16 1
k
kk ( )
= k kk
( )2 11
... e EikH
k
kk
k
2
1
1 2 2 2 2 12
( ) log!
... 612
4 2 22
2 2
0
1
log log arccos logx x dx
... 2 2 3 ( ) ( )
=2
1 21
H
k kk
k ( )
=
log
( )
2
21
2
20
2
01 1 2
x
x xdx
x dx
e
x dx
e e ex x x x
... 8
1 .88580651860617413628...
2 4
1)(5
8
5
4
2log15
4
5
kk
k k
... 3
35
5 .8860710587430771455... 16
4 1e
( , )
... 5
.88620734825952123389... 2
31 3
2 1
1 2 1
0
erfk k
k k
k
( )
!( )
.88622692545275801365... 2
AS 6.4.2
=( ) ( )!
!
1 31 12
1
k k
k
k
k
= 3
2
AS 6.1.9
= sin2 2
20
x
xdx
= e d LY 6.29 xx
2
0
= ex
xdxxtan sin
cos
2
20
GR 3.963.1
... 24 5 11
4
0
( )
x
e edxx x
... 24 5 11
44
0
1
( )log( )
x
xdx
... 2 21
2log
... 5
2
2
22
2
22 2 12
2 1
cot ( )
kk
k
k
k
... ( )
11
1
k
kk k
... 5
3
... 1
2
2
3 3
1
3 120
csch ( )k
k k
... sech1
2
2
2 41 2 1 22
0
e e
E
kk
kk
/ / ( )! AS 4.5.66
...
log log
1 3
2
1 3
2
i i
= log ( )( )
11
13
31
1
1
k
k
kk
k
k
... log
( )
4
2 1
k
k kk
... 9 3log
... ( )
1
3 2
1
1
k
kk
... F
kk
k
2
1 !
.8876841582354967259... 2 21
2
1
1 2 10
log ( )( )
( )! (Ei
k k
k
kk )
... k kk
5 2
2
1 ( )
... 3 3
6
CFG D14
8 .8881944173155888501... 79
... ( )
1
5 10
k
k k
=8
9
1
80
( )k
kk
... 253
5
1
1 5
1
1 52
2 1
2
1
21
( )
( / )
( )
( / )
k k
k k k
... 1
1 11 k kk ! ( )
... 11
2 41
k k kk ( )( )
...
41
2 13
2
0
Lix dx
ex / 2
... 2 3 3
81 13 20
log
( )
x
xdx
... 15
8
...
2 2
3
23
5
6log log
... 1
42 3
1
2 2
2
21
, ,log x
xdx
.890898718140339304740... 2 11
6 11 6
1
/ ( )k
k k
.8912127981113023761... HypPFQx dx
ex[{ , , },{ , , }, ]log
111 2 2 2 11
2
2
0
1
=( )
!( )
1
1 30
k
k k k
... 3
8
5
2 1
4
0
2
x dx
ex
...
41 2 2
0
2
e dcos (tan)/
x GR 3.716.10
= dxx
x
02
2
1
cos
.891809...
1 )1(
)(
k kk
k
... dx
x x21 1log( )
... 45
4
1
4 1 10 0
log( )
( ) /
k
kk
xk
dx
e 4
... 1 3
1 2
/
/
... 3 23log
...
0
2/
2! k
ke
k
ee
... 11
2 kpp
p primep primek
( )
... ii eLieLi 24
24
... 4
3
1
3
1
3
3
0
e dxx
... HypPFQk
k
k
13
2
3
2
1
4
1
2 1 20
, , ,( )
(( )!!)
=
210
0
2
H x( ) sin(cos )/
dx
... x x
edxx
2
0
1
1
log( )
...
1
2
)2(4
360
7
k
k
k
H
= dxx
xx
1
0
2log)1log(
... 22
2cos ( )/ ( e i ii ii i
)
... 82
1
96
1
414 3
1 42
33
2
41
Gk
kk
( ) ( )( / )
.8944271909999158786... 2
5
1
16
2
0
( )k
kk
k
k
=17
19
.89493406684822643647...
2
22 26
3
4
2 1
11 1
k
k kk k
k
k
k( )( ) ( ( ) )
= 1 122 3 2
2 2k k k kk k
k k
( ) ( )
= 2
2
1
1 13 31
2
22k k
k k
k k kk k
( ) ( )
1 .89493406684822643647... 2 2
0
1
6
1
4
1
1
( ) logx x x
xdx
.895105505200094223399... ( )
( )!
/k
k
e
kk
k
k
1 1
11
1
22
1 .89511781635593675547... Ei li ek kk
( ) ( )!
11
1
AS 5.1.10
.89533362517016905663... si
k k
k k
k
( ) ( )
( )!(
2
2
1 2
2 1 2 10
)
.89580523867937996258... iLi e Li e
k
ki i
k2 4 41
( ) ( )sin
4
... 2
4 2
1
2csc
= ( )
1
2 1
1
21
k
k k
.89601893592680657945... sin
2 21
1
16 121
kk
J1064
.89682841384729240489...
2
1
2
1
22 12Li , ,
= log log log log2 222 2 2 3 3 2
1
3
Li
=( )
( )
1
2 1 20
120
k
kk
xk
x
e
... ( )
!
/k
k
e
kkk
k
k
3
20
1 2
31
... 2
11
... 7 3 1
6 2 1 12 70
dx
x /
=115
128
1
53 0
5
3
1
, ,k
kk
... 1
2 1
2
1
log
( )
k
k kk
.8989794855663561964... 2 6 41
8 1
2
0
( )
( )
k
kk k
k
k
4 .8989794855663561964... 24 2 6
.89917234483108405356... 3
2
1
3
1
3 2 10
erf
k k
k
kk
( )
! ( )
.89918040670820836708... 22 2 2
1
2 121
cot
/kk 4
.89920527550843900193... 3
4 17
17
2
1
3 221
tan
k kk
9 .8994949366116653416... 98
... 1
1 G
.90000000000000000000 = 9
10
1
90
( )k
kk
.9003163161571060696... 2 2
11
16 21
kk
GR 1.431
= 0
1 4/
= cos
2 21
kk
GR 1.439.2
2 .900562693409322466279... 3 3
16 24 22
22
0
1 ( )log log arccot x x dx
1 .90071498596850182787... 2
1
2
2
( )kk
k
2 .90102238777699398997... 1
1 21 kk ! /
... 11 11
5
1
ek
kk
( !
...
0
2
32
)1(2
2
12
kk
k
k
kK
... 1
2
2
42
0
1
si
x x d( )
log cos x
... 3 3
43 1 3 0
1 1
31
( )( ) , ,
( )
k
k k AS 23.2.19
= log( )log
10
1
xx
xdx GR 4.315.1
= x dx
ex
5
0
2
1
.901644258527509671814... 2 10
11
3
4
3
1
2
1
2 3 1F
k
k
kk
, , ,( )
( )
... Brun’s constant, the sum of the reciprocals of the twin primes: primeatwinp p
1
Proven by Brun in 1919 to converge even though it is still unknown whether the number of non-zero terms is finite.
... ( )( )
11
21
2
1
kk
k
k k
... cos 2
... 5
3
... ( ) ( )
1 13
2
k
k
k
... sin Ime
ie2
...
( )
( )
k
kk
3
12
... 6
3
62 3
1
6 10
log( )k
k k
... 1
3 10k
k
1 .9041487926756249296...
3
2
3
2
3
2
3
4,2,
2
3F
3
210
3/211 IIe
=
k
k
k
k
kk
2
3!0
= 6463
512
1
55 0
5
5
1
, ,k
kk
... 3
82
7 3
4
16 6 1
2 2 1
2 2
3
log( ) k k
k kk
=( ) ( )
1
2
2
2
k
kk
k k
... ( )
( )!( )
1
2 4 10
k
k k k J974
= 2
2 12
0
1
21
2C x dxx
dx
x
cos cos
... e
e
3
3
1
1
J148
... 4108
243
1
45 0
4
5
1
, ,k
kk
... ( )k
Fkk
1
12
... e
.90575727880447613109... si
k k k
k
k
( ) ( )
!( )( )
2 1
2 1120
... 2 2 22 1
11
1 22
1
1
log log( )
( )
( )(
kH
k
H
k kk
kk
kk
)
=
12 24k
k
kk
H
... 1
2 2 11k
k k (
... 3
53 3 3 4 2
1
1 1 60
log log( )( /k kk )
... e
3
... 5
4
1
4
1
4
4
0
e dxx
... 1
1 k kk !
... 2 1
21
2
1
22 1
1
16 1
2F
k
k
k
k
k, , ,( )
( )
.9066882461958017498...
2 0
log sinx x
xdx
.9068996821171089253...
2 3
1
6 1
1
6 50
k kk
J84, K ex. 109e
= maximum packing density of disks CFG D10
= ( )
( )(
1
1 3 10
k
k k k )
= ( )
( )
1
3 2 10
k
kk k
J273
=dx
x
dx
x x
dx
x20
4 20
203 1
3 1
= x dx
x x
2
4 20 1
= xx
dx2
0
1
311
log
1 .90709580309488459886...
1
2)12(
31
2
3cosh
4
1
k k
.90749909976283678172... b k
k
k
kk
k k
( )( )
( )log ( )2
1 1
2 2
Titchmarsh 1.6.3
.90751894316228530929... ( ) logk kk
12
23 .90778787385011353615... 5
64
5 4
0
x
e edxx x
= GR 4.227.3 log tan/
x dx4
0
2
=log4
20
1
1
x
xdx
GR 4.263.2
... Lii
Lii
2 2
1
2
1
2
.90812931549667023319... 4 2
4190 48 96
1
768
2
cos( / )k
kk
GR 1.443.6
= 1
2 42
42Li e Li ei i/ /
.90819273744789092436... 1
32
5 3
2
5 3
2
i i
= k
kk k
k k
1
13 1 3 23
2 1
( ) ( )
.90857672552682615638... ( ) ( )
( / )1 2
31
4
3
1
6
4
3 1
k
kk 3
.90861473697907307078... 0
1 1 112
1 2 1
2
( ) log( )k
k kkk
k
kjk
kj
jk
... ( ) ( )
1 2 22
2
k
k
= 10
11
1
100
( )k
kk
... 1
2 k kk !log
... sin( )
( )!
sin( / )
!2
1 2
2 1
2 22 1
0 1
k k
k
k
kk
k
k
AS 4.3.65
... 1 1 1
2 1
ex dx
e(cos sin )sinlog
... 3 3 2 2 12
0
1
log log log
x
dx
... 5
23 5
1
5 1
2 2
0
kk k
k
k( )
... 6
1 .91002687845029058832... 1 2 35
44 1
31
( ) ( ) ( )H
kk
k
.91011092585744407479... ( )2
2
1
221
2 21
k
kLi
kkk k
.91023922662683739361... 1
3 30log
dx
x J153
... 1
8 42
1
420
cothkk
.91059849921261470706... sin(log ) Im i
.91060585540584174737... Li3
4
5
1 .910686134642447997691... 1
2 2
1
2
2
21
eerf
k k
k
k
k
!
( )!
244 .91078944598913500831... 8
396 64 2
33 2
22
0
log ( )/x Li x dx
.911324776360696926457...
2
1)()(k
kk
.91189065278103994299... 9
2
.91194931412646529928... 3
34
23 .9121636761437509037... 65
5 1e
( , )
... 1
11219 14 2 2 2 csc( )
... 73
...
121 k
kk
...
( )32
... 5
22 CGF D4
.9142425426232080819... 7
2
7
1
7 220
arctandx
x x
=32
354 1 2
1
4 40
( , / )( )k
k k
2 .91457744017592816073... cosh( )
33
20
k
k k !
... 2
1cot
2
1
=
i e
e
i i
i
B
k
i
i
kk
k
1
2 1 2
1 1 1
1 1 1
1
22
0
cos sin
cos sin
( )
( )!
= limsin
aa
k
k
k
0 1
.91547952683760158139... ( )
1
7 10
k
k k
... 120 46 1255
4
1
( )!( )
ek
k kk
...
32 3
1
1 320
tanh/
k kk
.91596559417721901505... G ( ) Im ( )2 2Li i , Catalan’s constant LY 6.75
=
2
1
2
1
28
2log22
iLi
iLi
i
= ( )
( )
1
2 1 20
k
k k
= 1
4 3
1
4 12 21 ( ) ( )k kk
= 1
2
4
2 2 1
2
20
( !)
( )!( )
k
k k
k
k
Adamchik (23)
= 1
16
3 1 1
42
1
( )( )( )
k
kk
kk
Adamchik (27)
= 1
8 21
3
41
kkk
k
, Adamchik (28)
= 8
2 33
8
12
2 1 20
log
( )
k
kk
k
= x
e edxx x
0
=
log logx
xdx
x
xdx2
0
1
211 1
= log1
1 10
1
2
x
x
dx
x GR 4.297.4
=
log
1
120
1 x
x
dx
x Adamchik (12)
=
log1
2 1
2
20
1 x dx
x Adamchik (13)
= GR 4.227.4 log tan/
x dx0
4
= 1
2 0
2 x
xdx
sin
/
Adamchik (2)
= 1
2 0
x x dsech
x Adamchik (4)
= ar Adamchik (18) csinh(sin )/
x dx0
2
= Adamchik (5), (6) 2 2 2 20
4
0
4
log( sin ) log( cos )/ /
x dx x d
= arctan arctanx
xdx
x
xdx
0
1
1
GR 4.531.1
=
arctan x
xdx
1 20
= 1
22
0
1
K x dx( ) Adamchik (16)
= 1
22
0
1
E x dx( ) Adamchik (17)
= 1
1 10
1
0
1
( )x y x ydx dy
Adamchik (18)
.9159981926890739016... ( log ) log( )
( )
8 3 18 2
12 2
1
1
4
4 40
x
x xdx
5 .91607978309961604256... 35
... Li13
5
.91666666666666666666 = 11
12
1
110
( )k
kk
... cos cosh( )
( )!2 2
1 2
44 4
0
k k
k k
... HypPFQk
k
k
k
k
, , , ,( )
( )!
1
4
3
41
1
64
1
4
2
0
... 2 4 ( )
.91775980474025243402...
93 3 1 3
1
1
3
3 30
(log )log( )
( )
x
x xdx
... 6
5
... 21
2 2
1
2
1
2 1 21 4 1 20
sin( )
( )!/ /
J
k
k
kk
... 7
16
13
32
1
2 21 4
2
0
e erfk k
kk
/ !
( )
!
... log lim!
/2 1 2
n
n
n
n e
n
= Stirling’s constant , GKP 9.99
the constant in ...12
1
2
loglog!log
n
nnnnn
= GR 6.441.2 log ( ) x dx0
1
=1
1 20
1
log logx
x
x
x dx
x x
GR 4.283.3
.9190194775937444302... sinh
exp( )
41
1 44
2 1
k
k
kk
1
= 1
2 2 2 i i
... 7
4
... ( ) ( )( ) ( )1 1i i
... H
kk
k
2
1 !
.9193953882637205652... 2 2 1 41
2
1
2 12
1
cos sin( )
( )!
k
k k k
... 12
2 1
2
20
sin
sin
x
xdx
.920673594207792318945... e
e e
k
ekk( )
, ,
1
11 02
0
=( )k
ekk
11
... ( ) !
( )! /
1
2 1 2
2 1
2 201 8
k
kk
k
k eerfi
... log log( )2
22
1
1 20
1
G
x
xdx GR 4.292.1
= arccosx
xdx
10
1
GR 4.521.2
= x x
xdx
sin
cos
/
10
2
GR 3.791.9
12 .92111120529372434463... 2 82
1
1 420
csch ( )
/
k
k k
... 2
2 2
2 2 2 2
sin sinh
cosh cos
= 11
11 1 44
2
1
1
k
kk
k
k
( ) ( ) 1
... 4 21 2
2
1
log( /
H
k kk
k )
7 .9219918406294940337... e I
k
k
kk
20
0
2
2
1
2
1
1
2 1( )
( )!
... 2
216 4
1
16
2
cos( / )k
kk
= 1
2 22
22Li e Li ei i/ /
... erfk kk
k
1
2
1
4 2 1
1
0
( )
! ( )
... ( )x d 10
1
x
... 3
23 ( )
... 2
1
2 1( )kk
=12
13
1
120
( )k
kk
... ( ) ( )
( )!sin
1 4
2 1
1 11
2 211
k
kk
k
k k
k
...
4 1
4 1
2
12
0
tanhk kk
.923879532511286756128... 8
3sin
8cos
4
22
... 90 1
44 41
( )
( )k
kk
... point at which Dawson’s integral, , attains its e e dx t
x
2 2
0
t
maximum of 0.5410442246. AS 7.1.17
.92419624074659374589... 4 2
31 2
1
27 331
log ( )
k
k k k Berndt 2.5.4
= Hk
kk
/2
1 2
... ( )
log
1
2
k
k k
... 2
5025 5 5 1
cot csclog
x
xdx
... ( )
1
8 10
k
k k
5 .924696858532122437317...
2 3
1)(5
36
5
2
3log5
3
5
kk
k k
...
1
32
22
1
22
)14(
)14(41)2(
432
3
kkk k
kkk
k
= dxx
x
121
arctan
... erfik
k kk
( )( )
!( )!1 2
12
121
!
... sec1
2
1
e ei i
... 1
4 1
1
23 1
1
21
1
2( )( ) cosh ( ) sinh
ee e
sech
=( )
1
12 20
k
k k
... 11 12
2 21
sinh ( ) ( )k i k ik
2
=2 2
2 1
2
4 21
k
k kk
... 3 2 5 ( )
... 12
2
cosh
... 2
2128
22
4 1
2 2 1
log( )
( )
k k k
k kkkk
... 1
122
2 12 2
3
0
loglog
( )( )
x
x xdx GR 4.262.3
... 1
8
1
4
1
4
1
2
5
41
12
2 1 41
Fdx
x, , ,
.927099379463425416951... 3 3
26 2 3 6 2 3
1
4
22
2
loglog log log
Li
=( )
( )
1
3 1 20
130
k
kk
xk
x
e
...
1
5 3 5
10
5 5
2
5 3 5
10
5 5
2
= 2 1
11 12
21
2
k
k k kF k
k
kk
k
( )
( ) ( )
... 21
21
21
2
1
4 2 10
arctan log log( )
( )
ii
ii
k
k
kk
... 7
4
2
2 ( )
... 1
2
23
23
23
1
Li e Li ek
ki i
k
/ / cos( / )
...
4
41105
8
4
( )
( )
( )k
kk
HW Thm. 300
2 .92795159370697612701... e2 1 2 1 2 1 2 1 2 1cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )
=
ie e
k
ke e
k
k
i i
2
22 2
1
sin
!
1 .92801312657238221592... 2 1 2 14
20
( log ) log log
xx dx GR 4.222.3
3 .928082040192025996... pd k
kk
( )
!
1
.92820323027550917411... 4 3 61
12 1
2
0
( )
( )
k
kk k
k
k
6 .92820323027550917411... 48 4 32
61
2
k
k
k
kk
13
14
1
130
( )k
kk
... 2 3
0
1
3
49
36
1
1
( ) logx x
xdx
... 1
22 2
2
220
Ik k
k
k
( )!( )!
... 5
2 2
1
1
55
51
i e
e
k
k
i
ik
logsin
... e
... ( ) ( ) ( )2 3 4 312
42
k k
kk
... ( ) ( ) ( )2 3 412
41
k k
kk
... ( )
!/3
11
1
1
3k
ke
k
k
k
... 3 3 1 19
20
log log log
xx
dx GR 4.222.3
... 2 10
11
3
4
3
1
3
1
3 3 1F
k
k
kk
, , ,( )
( )
... 263
2310 2 1515
csc
... e
... ( )
!( )!
k
k k kI
k kk k
1
12
1 1
21
1
.93122985945271217726... 21
2
1
8 2 1
2
0
arcsinh
( )
( )
k
kk k
k
k
.93137635638313358185... | ( )| k
kk 21
6 .93147180559945309417... 10 2log
.9314760947318097375...
( )( )
3360 1
4
31
k H
kk
k
1 .9316012105732472119... e e k
k e
e
kk
2
2 0
cosh
!
... 3 4
32 2 14 20
log
( )
x
xdx
... ( )
1
9 10
k
k k
...
3
2 3 21
log/
k
kk
.9326813147863510178... 4
21
1
162 41
sinh
kk
.932831259045789401392...
1
02
22
123
12
)(2
)1(2log2
12 x
dxxLi
kkk
k
1 .932910130264518932837... 37
163
12
2
22
0
( ) ( )x Li x dx
... cos( )
( )!
1 1
2 20e k
k
kk
e
AS 4.3.66, LY 6.110
... 8777
1
ek
kk
!
=14
15
1
140
( )k
kk
5 .9336673104466320167... 1
1256
1
4
3
4 13 3
3
21
( ) ( ) log
x dx
x
...
1
1
... 11 1
1
ee
ke
k k
k
( )
!
... 413 1 10
0e
k
k
k
k
( )
!
.93478807021696950745... 2 7
6
4
3
12
13
161
k k
k kk
( )
( )( ) J1061
.93480220054467930941...
2
3
)12(!)!12(
!)!2(4
2)1(
12
2
k kkk
k
=1
1 2
1 1
2
4
2 121
22
21( / )
( ) ( ) ( )
( )k
k k
kk
k
kk k
1 .93480220054467930941... dzdydxxyz
zyx
1
0
1
0
1
0
2
13
2
2 .934802200544679309417... 2 2
022
1
x x d
x
sin
cos
x
4 .93480220054467930941...
2
1
2
1
2
( ) logi i
= 3 21
21
212
20
( )( )
,
kk
=
211 212 )!2(
4)!1()!1(2
4
!)!12(
!)!2(
k
k
k
k
k k
kk
kk
kkk
k
= k k kk k
k kk k
2
2
2
2 22
14 1
1 ( ) ( )
( )
= 2 12arcsin
= log( )
2
02
1
1 1
x
x
dx
x x GR 4.297.5
= logtan
tan2
0
1
1
x
dx
x GR 4.323.3
8 .93480220054467930941...
2
1
24
1
2
( )
... 7
7291
1
3 1
1
3 1
4
4 41
4
( )
( )
( )
( )
k k
k k k
... 5
2 25 5
0
1 51
1
25 21
/ kk
... ii eLieLi 2
32
3
... log( )
( )(2
3 2
1
2 1 3 10
k
k k k )
... 11
12
( ) ( )k kk
.936342294367604449... sin1
11
12 2
2
kk2
... x dx
e ex x
3
1
.93709560427462468743...
81 2 2 2
0
2
( log ) log(sin )cos/
x x dx GR 4.384.10
= 1 2
0
1
x x dlog x GR 4.241.9
= xe e dxx x
1 2
0
GR 3.431.2
7 .9372539331937717715... 63 3 7
... 3
2 2 6
2
2 11
2 2
1
log( )
k
kk
k
Admchick-Srivastava 2.23
=15
16
1
150
( )k
kk
... ( )
1 1
1
k
kk k
k
...
( )log
2 21
k
kk
Berndt 8.23.8
= 1
1 311 ( )!
log
n
k
k
n
kn
... ( )
1 1
12
k
kk k
1 .93789229251873876097... 3 2
2 20
2
2016 1
log
( )
x dx
x
x dx
e ex x
= i Li i Li i3 3( ) ( )
= log log2
20
1 2
211 1
x dx
x
x dx
x
GR 4.261.6
=
log2
40 1
x dx
x
= GR 4.227.7 log tan/
x dx2
0
4
= dydxyx
yx
1
0
1
022
22
1
)log(
.93795458450141751816... )()(31524
1
1222
24
iiik
H
k
k
.93809428703288482665... ( )
1
10 10
k
k k
... 81
24
1
28
1
2 4 10
cos sin( )
( )! ( )
k
kk k k
... Jk
k
kk
1 20
1
2
1
16
( )
( !)
... ( ) ( )
( )! /
1 2
1
11
12 1
12
k
kk
k
k
k k e
... k Hk
kk
2 3
1 2
( )
... 12
1 1( ) ( )k kk
...
2
coth
... 5 5
4
5 5
5 5
25 5
4
5
21
2
5log
log
=5
5 1
1
1 4 5 51 02
2k k k k
k
k kk
k( ) ( )( / )
( )
4 .939030025676363443... 3 24
256 2 23040
12 4
41
( )
( )
( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
... 3
33
1 .9395976608404063362... log log21
2
2 2
2 2 22
1
H kk
k
4 .9395976608404063362... 3 21
2
2 2
2 2 22 1
1
log logkH k
kk
...
primekthp
k
k
pk 1)()(
...
primep pp
p42
22
1
1
)4(
)6(
21
2
... J310 g4
.94015085153961392796... ( )
( )
k
k kk
12 !
... ( )
!
/2 1
1
1
1
2
k
k
e
k
1
k
k
k
... 115
384
5
50
sin x
x GR 3.827.11
... 24 2 321
8
2 2
0
kk
k
k
=16
17
1
160
( )k
kk
... 89
8
1
8 10
log( )
( )
k
kk k
... 42 2
3 CFG D4
.94286923678411146019... 2
316 4
1
12
sin k
kk
Davis 3.32
= iLi e Li ei i
2 3 3
.942989417989417 = 7129
7560
1
3 31
k kk ( / )
.94308256800936130684... HypPFQ[{ , , , },{ , , , }, ]1 1 1 1 2 2 2 2 1
=( )
!( )
1
1 40
k
k k k
... 5
4
2
42
1
222
cot
kk
= 2 21
1
k
k
k
( ) 1
... 3 2 66
1
1 620
csch ( )
/
k
k k
.94409328404076973180... ( ) arcsin arcsin3
4
2 3
0
1
x x
xdx
.9442719099991587856... 4 5 81
16 1
2
0
( )
( )
k
kk k
k
k
8 .9442719099991587856... 80 4 5
=17
18
1
170
( )k
kk
... cos( )
( )!
1
3
1
2 9
k
kk AS 4.3.66, LY 6.110
...
21
2
1
2 13 30
Lik
k
kk
( )
( )
=log2
21
2
02 2
x
x xdx
x
edxx
1
= log2
0
1
2
x
xdx
.94529965343349825190... 0
1 1
( )
( )
k
kk
!
.94530872048294188123... 8
15
5
2
1 .9455890776221095451... ( )
( !)
2 212
10
1
k
kI
kk k
... log76
71
Li
.94608307036718301494... 1
0
sin)1( dx
x
xsi
=( )
( )!(
1
2 1 2 10
k
k k k ) AS 5.2.14
= sin1
1 x
dx
x
= 1
0
coslog dxxx
... 320
epf k k
kk
( )
... sink
kk 21
...
52 5
1
1 520
tan/
k kk
... 1
43 1 31 3
3
1
e ek
kk
cos cossin(sin ) sin(sin )sin
!
= ie e e ee e e ei i i
83 3
3 3
i
... 7
7204
7 4
8
14 1
41
( )( ) ( )k
k k J306
.947123885654706166761...
2
1
42
1
23 41 4
1 4//
/cot ( ) coth( )
i i
= k
kk
2
41 2
.94716928635947485958... 11
11
2
1
321
tanh
k kk
... 18
19
1
180
( )k
kk
...
162 8 2 82 2 2 log log 2
dx
= GR 4.335.2e xx
2 2
0
log
...
sin1
113 2
1
kk
... 8
1 21
8 7
1
8 11
k kk
J78, J264
... Q k
kk
( )
!
1
... ( ) ( )2 3
3
4 .949100893673262820934... 1
3 1 ( )
.94939747687277389162... arctan(sinh csc ) ( ) arctan1 1 11
11
k
k k
[Ramanujan] Berndt Ch. 2, Eq. 11.4
.9494332401401025766... k k
kk
kk3 1 31
1
1
( )
... 2
6
.94949832897257497482... 2 1 1
4120
sin ( )
!( )
k
kk k k
... 2
2 216 3
11
6 1
1
6 1
( )
( )
( )
( )
k k
k k k J329
=
log x
xdx6
0 1
9 .9498743710661995473... 99
=19
20
1
190
( )k
kk
... 2
22
206
31 5
2
1
2 2 1
log( ) ( !)
( )!( )
k
k
k
k k
...
6
3
2
1
3
1
3 221
coth/
kk
.95075128294937996421...
1
3
)2(
)3(7
8
k k
k
.95105651629515357212... sin2
5
.9511130618309... ( )( )
1 14
1
k
k
k
k
1 .95136312812584743609... 2 21 8
22
1
1
sin( )
( )!
k k
k k
... 15
8
2
6
2
0
e
k k
k
k !( )
... 1
36
1
6
2
3
1
3 11 1
20
( ) ( ) ( )
( )
k
k k
=
log logx
xdx
x x
x
x dx
e ex x1 130
1
31
20
.9520569031595942854... ( )31
4
1 12 5
2
k k k kk
3
= ( ) ( )k kk
2 12
.95217131705368432233... 2 2
2 2 0
4
logcos log(sin )
/
x x
dx
.952380952380952380 = 20
21
1
200
( )k
kk
... 1
4
1
22
1
2
1
2 2 1 20
, ,( )
( )
k
kk k
2 .95249244201255975651... ( )! !
ej kkj
112
11
.95257412682243321912... e
eek k
3
33
1
1
21
3
21
tanh ( ) J944
...
2
1
1 220e t
tdt
cos
... dxx
x
0
224
2cos2log
1
2
)4(11
180
11
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
... 551
2101
3
5
0
ek
kk ( )!
.954085357876006213145... ( )
!
k
kk
2
... Eik k
k
k
( ) log!
2 22
1
AS 5.1.10
... 1 4 5 ( ) ( )
... )3()2(23
)3(2
... 3
23 2 1
1
20
1
log log log
x
dx
... 2 100
2
I e x( ) sin dx
.95492965855137201461... 3
11
36 21
kk
GR 1.431
= cos
6 21
kk
GR 1.439.1
= 0
1 6/
2 .95555688550730281453... !
1)(2
2 k
k
k
k
k
... 7
128
3 3
42 1 13
1
( )( )k k
k
=k k k
kk
( )
( )
4 2
2 42
4 1
1
... 2931587
2 3
6
0
ek
kk
( )!
... 5
162
33
31
sin /k
kk
GR 1.443.5
.95706091455937067489... G
... log ( ) snns
12
.957657554360285763750... 1
22
1
2
1
2 11
cot tank kk
Berndt ch. 31
...
0
2
2
2
)1log(
2
1arcsinh
x
x
2980 .95798704172827474359... e8
...
22 2
2
1
21 1
1
81 1 ( ) ( ) ( ) ( )( ) ( )i i i i
5
81 1 21 1i
i i ( ) ( )( ) ( ) (3 )
1
= ( ) ( ) ( )
1 22
2
k
k
k k k
.95833608906520793715... ( )
( )!
1 1
21
k
k k
.95835813283300701621... cos cosh( )
( )!
1
2
1
2
1
40
k
k k
.95838045456309456205... 1
1024
5
8
3
82
1
82
8 82 2 2 3 2
( ) ( ) ( ) cot csc
= 11
4 1
1
4 13 31
( )
( )
( )
( )
k k
k k k
.95853147061909644370... 1
3
3
2 3
1 1
3 10
cos( )
( )
e k
k
k !
... 1 2
21
2
2 20
tanh( / )( )
/
/ /
e
e eek k
k
J944
2 .95867511918863889231...
1
43
4
... 21
2
1
2 1 41
1
402 2
1
sin( )
( )!
k
kk kk k
GR 1.431
= 0
1 2/
.95924696800225890378... 4 11
2 1
1
21
cik k
k
k
( )( )
( )!
... 1
2 1 2 11 2
0sinh( / )( / )
e
ee k
k
J942
.95974233961488293932...
03
3/2
233
2
x
dxx
.95985740794367609075... 2
12sin
1
2
1sin1cos22log
2
1cos)1(
2
1
1
k
kk
.9599364907945658556...
1
1)1()(k
kk
... k k
kk
( ( ) )
!
2
2
1
.96031092683032321590... 16
17
1
4
1
2 1 4
1 2
0
arcsinh
( ) ( !)
( )!
k
kk
k
k
.9603271579367892680... 4
4 21
2
1
1 4 2 11 40e
erfk k
k
kk
/
( )
( )! (
)
... 32 2
151
1
16 22
kk
... 2 22 2
0
2
48 8
2
4
2
4
log loglogsin cos
/
x x dx
... 2 22 1
2
1
log( )
H
k kk
k
... ( ) ( )
1
2 1
1
1
k
kk
k
... 2 21
2 2
1
8 2 10
arctan( )
( )
k
kk k
... 3 3 2 ( ) ( )
... 4 3
5
( )
... ( )
( )!
2 1
113
1
2
k
k
e
kk
k
k
... 44
5
1
2
1
4 2 1 10
log arctan( )
( )(
k
kk k k )
... 21
22
1 5
2
1
16 2 1
2
0
arcsinh
log( )
( )
k
kk k
k
k
... log 7 22 3
1
33 3
arccot
= 11
3
1
6 1
1
6 13 11
( )k
kk k k
... 2 5 ( )
... 5 1 1 1 11 5 2 5 3 5 4 5 1 ( ) ( ) ( ) ( )/ / /
/
= 11
52
kk
... 5
8
...
2 2 1
1
2
1 2
1 21log
( / )
( / )
=
3
21
.96400656328619110796... ( log )log( )
( )1 2
1
1
2
2 20
x
x xdx
... tanh22 2
2 2
e e
e e
3 .9640474536922083937... ( ) ( ) ( ) (i i i )i 1 1
.9641198407218830887... 26
4 23 3
4
1
2
2 1
4 31
log( ) ( )k
k k k
... 1
5 51
( )
( )
k
kk
...
( )
( )
( )10
5 51
k
kk
HW Thm. 300
.96558322350755543793... 1
2 20k
k
...
( )2
2 12 11
kk
k
... 2
41 3
5
12 sin
AS 4.3.46
... 6 5 3
108 16 20
log
( )
x
xdx
... erfk k
k k
k
3
2
2 1 3
2 2 1
2 1
2 1
( )
! ( )
... ( ) ( ) ( ) ( )2 3 4 5
... 1
4
1
23
1
2 1
2
21
, ,log
/
x
xdx
2
... 1
21 1 1 1
0
1
cosh sin cos sinh cos cosh x x dx
... 11
1
kk !!
...
1
0
1
0
1
0
1
02222
3
12log24
8dzdydxdw
zyxw
zyxwG
.96676638530855214796... 7
71
1
49 21
sin
kk
GR 1.431
= 0
1 7/
... ( )( ) ( )
11
212
2
k
k
kk k
.96761269589907614401... ( )4 3
2 14 31
kk
k
.9681192775644117... 5
.96854609799918165608... 1
64
1
8
5
8
1
4 11 1
20
( ) ( ) ( )
( )
k
k k
=
log logx
xdx
x x
xdx
1 140
1 2
41
= x x
x xdx
log3 1
1
3 .96874800690391485217... F Hk k
kk 21
4 .96888875288688326456... 2
2 2
( )
( )
k
kk
!
.96891242171064478414... cos( )
( )!
1
4
1
2 160
k
kk k
.96894614625936938048...
3
3032
31
2 1
( )( )
( )
k
k k AS 23.2.21
= Im ( ) ( ) ( )Li ii
Li i Li i3 32 3
= sin /k
kk
23
1
=
1
0
1
0
1
02221 zyx
dzdydx
... 2
2 11
k
k k k! ( )
Titchmarsh 14.32.3
.96936462317800478923...
2 2 2 121
2
csc
=( )
1
2
1
4 21
k
k k k
... 4
729 3
5
5
g J310
... 2
... 2 2
... G1 3/
=43867
798 9 B
...
2
0
2
0 sin2
sin
cos2
cos1
3
22
x
dxxdx
x
x
... k k kk
( ) ( )
1 12
... 15 5
165
11 5 0
1
51
( )( )
( )( , , )
k
k k AS 23.2.19
... k k kk
( ) ( )
1 12
... 276 150 22
4
1
logk Hk
kk
... 15 5
16
1
1 50
( ) ( )
( )
k
k k
... 61
6
1
6 2 10
sin( )
( )
k
kk k !
= 11
6 2 21
kk
... log7
2
= 11
3 6 1
1
3 6 13 1 31
( )
( )
( )
( )
k
k
k
kk k k
K ex. 108
1 .97318197252508458260... 1
216
2
3
1
62 2
2
20
( ) ( )
x dx
e ex x
= ( )( ) ( ) ( ) (/ / / /3
2
2
31 1 11 3
32 3 2 3
3
1 3 Li Li 1
= log log log2
30
1 2
2 11
2
311 1
x
xdx
x
x xdx
x x
xdx
.97336024835078271547...
1
3
1
2
2
3
2
32 3( ) /
2 .97356555791412288969... 4 2 3 3 ( ) ( )
.97402386878476711062... S k k
kk
2
1
2
2
( , )
( )!
.97407698418010668087... log( )1 e
... dxexx
x
e
0
22
))(1(
log
33
1
.9744447165890447142... 1
4
1
42
1
2
1
4 2 1 2 10
, ,( )
( )( )
k
kk k k
.97449535840443264512... 8
8 2 31
sin cos
kk
GR 1.439.1
.97499098879872209672... 1
22 1 2
1
3 40
log( )
/
k
k k
= dx
x x( ) cosh /1 420
GR 3.527.10
... 3
64
2
16
2
16 4
4 2 2 2
2 20
log log log x
x xdx
... sin Im ( )
2
2i
1 .97553189198547105414... 2 21 16
22
1
1
sin( )
( )!
k k
k k
... 4
1524 4
2
4 4
0
( )x
e ex x
... I HypPFQ0 12
13
2
3
2
1
4( ) { }, , ,
= e dxsin
0
1
x
... 11
360
3
24
1
22
1
42
2
21
( ) ( )( )
H
kk
k
18 MI 4, p. 15
.97645773647066825248... sin( )
( )!(
1
2
1
2 1 2 11
1
2 11
kk
k
kk k
)
...
2
2 211
1
21
1
2 1
cos( )kk
2
... J314 h2
1 .9773043502972961182... ( ) ( )
( ) ( ) ( )2 3
3
6
21
31
12
1
k
k
k
kk k
HW Thm. 290
...
I e dx ex x0
0
2
0
1( ) cos cos dx
2 .9775435572485657664...
12 24
2 22
2
2 2 2
log log
log
= log sin2 2
20
x x
xdx
4 .9777855916963031499... e I Ik
k
k
kkk
0 1 1 10
1 13
22 2
2
2( ) ( ) ( , , )
!
F
... 5535
2
22
3
0
k
k
kk
k
... 1
100
1
10
3
5
1
5 11 1
20
( ) ( ) ( )
( )
k
k k
=x x
x xdx
log3 2
1
1 .9781119906559451108... 2
2 2
06
log xdx
ex GR 4.355.1
.97829490163210534585... G1 4/ ... 40320 9 18 ( ) ( )( )
... Li2
3
4
.97917286680253558830... cos cosh( )
( )!/ /
1
2
1
2
1
4 23 4 3 40
k
kk k
.97929648814722060913... 2 21
2 2
1
2 10
sin( )
( )!
k
kk k 8
.97933950487701827107... 161
4
1
2 1 4 12
0
sin( )
( )! (
k
kk k k )
.97987958188123309882... loglog( )1 3
2
1
1
2
20
1
x
xdx GR 4.295.38
.98025814346854719171... 2 21
2 2
2 1
32 2 10
arcsinh
k
k k
k
kk
( )
( )
... 31
3
1
9 2 10
sin( )
( )
k
kk k !
= 11
9 2 21
kk
.98174770424681038702... 5
16 1
5 2
x
xdx
/
GR 3.226.2
.98185090468537672707... ( )
!
( )
( )!!/2
2
2
21
1 1
1 2
1
2k
k
k
kek
k k
k
k
.98187215105020335672... iLi Li
21 12
3 42
1 4 ( ) ( )/ /
= sin / k
kk
42
1
... 6
5
2
5 5
3
5
3
5 1 5 30
csc /
dx
x
... 1 2
21
2
2 24
0
tanh( )
e
e eek k
k
J944
... 19
27 3
1 3 3
0
e k
k kk
/
!
... 22
9
2 2
3
2
3
2 2 12
2
30
log e
x x
e
x
dx
x
x x
GR 3.438.2
... 2 6 ( )
... ( )
( )
12
2 1 30
k
k k
kk
... 1 3
12
1
6 2
3
21
12 2
26
2
coshsin
ikk
... 945 1
66 61
( )
( )k
kk
...
sin1
114 2
1
kk
... 1
8
1
23
1
2 2
2
20
1
, ,log x
x
=i
Lii
Lii
2 23 3
2
.983194483680076021738... 691
675675
1
1
6
6
pp prime
... ( ) ( ) ( ) ( ) ( )2 3 4 5 6
.98361025039925204067... 2 16
2 21
2 1
3 1
31
Gk k
k
k
log( )
( )
.98372133768990415044... 1
144
1
12
7
12
1
6 11 1
20
( ) ( ) ( )
( )
k
k k
= x x
x xdx
log3 3
1
10 .98385007371209431669... 2 4 2 2 3 23 3
1
2
32
( ) ( )( )
k k
k kk
1
= ( ) ( )k k kk
2
2
1
.98419916936149897912... 403
393660
6
6
J312
... 2
636 5
3 ( ) ( )
.9845422648017221585... 2
22
20
1
62 2
3
4
1 1
log
( ) ( ) log ( )x x
xdx
.98481748453515847115... ( )
14
1
k
kk k
.98517143100941603869... 2 2 1 21 4 / , from elliptic integrals
... log1
3
.9855342964496961782... 96 16
15 4
24 4
1
( )
( )k
kk
.9855510912974351041... 31
302406
31 6
32
16 1
61
( )( ) ( )k
k k J306
... 4 23
8 8
11
43
40
logsin logsin( )
( )( )
k
k k k
... 1
512
5
8
1
8 12 2
2
40
1
( ) ( ) log
x
xdx
.98621483608613337832... 21
2
1
2 1 4 2 10
sik k
k
kk
( )
( )! ( )
6 .98632012366331278404... 5 9
4
2
3
2 2
0
e k
k k
k
k
!( )
.9865428606939705039... 11
768 21
1
4 1
1
4 1
4
41
( )
( )
( )
( )
k k
k k k 4 J327, J344
.98659098626254229130... 1
97210 3 351 3
1
432
2
3
1
63 2
( ) ( ) ( )2
= ( )
( )
1
3 1 30
k
k k
12 .98673982624599908550... 28402
2187
1
2
1 10
1
( )k
kk
k
.98696044010893586188... 2
10
... 1856
243
380
81
4
3 4
4
1
logk Hk
kk
... sin 2
.98787218579881572198... 4
105
7 15
42015 15
44 4
csc csch
= ( )
1
1542
k
k k
... 2
156
1
1
4 3
0
1
( ) logx x
xdx
...
2
3 12018 3 1 2 1
log x
xdx
.98861592946536921938... 3 23 1 1
1
144 21
kk
GR 1.431
= cos
3 2 21
kk
= 0
1 12/
... sin sinh ( )( )
1 1 12
4 2
2 1
0
!
kk
k k GR 1.413.1
= 114 4
1
kk
... 1
1536
1
4
3
43 3 ( ) ( )
= i
Li i Li i2 4 4( ) ( )
=( )
( )
sin /
1
2 1
24
04
1
k
k kk
k
k
1 .98905357643767967767... 3 2
609 3 1
log x dx
x
.9890559953279725554... 2 2 2
202 12
2
log ( )x dx
e x
...
( )log
22
21
k
kk
... 0
1 2( )kk
kk
.98958488339991993644... cos cosh( )
( )!
1
2
1
2
1
4 40
k
kk k
.98961583701809171839... 41
4
1
2 1 160
sin( )
( )!
k
kk k
= 11
16 2 21
kk
.98986584618905381178... 41
4
2 1
64 2 10
arcsinh
k
k k
k
kk
( )
( )
1 .98999249660044545727... 1 3 23
2
1 9
22
1
1
cos sin( )
( )!
k k
k k
.99004001943815994979... 1
3456
5
12
7
122
1
12
7 4 3
4322 2 2
3
( ) ( ) ( ) ( )
= 11
6 1
1
6 13 31
( )
( )
( )
( )
k k
k k k
... 10
3
74
81 3 2
3
1
kk
kk
78 .99044152834577217825... 1212 1
1
10
0
e
k
k
k
k
( )
( )!
.99076316985337943267... k kk
( )
12
2
.99101186342305209843... 1
41 1( ) ( ) ( ) ( )i i i i
21 .99114857512855266924... 7
4 .9914442354842448121... 6
3 ( )
.99166639109270971002... HypPFQkk
, , , , ,( )!
1
5
2
5
3
5
4
5
1
3125
1
50
.99171985583844431043... 1
7 71
( )
( )
k
kk
6 .99193429822870080627... ( )kk
2
7
.99213995900836... 7
.99220085376959424562... 4
125
2 53
31
sin /k
kk
GR 1.443.5
.99223652952251116935... 56
94815 3
7
7
g J310
1 .992294767124987392926...
1
2
30,2,
1
)1(
)1(
kke
k
ee
ee
... )()(768 55
5
iLiiLii
... 63 7
647
1 1
71
( )( )
( )
k
k k AS 23.2.19
... 2
1 1
0
( ) sin(tan )
e xdx
x GR 3.881.2
.99297974679457358056... ( )3 2
2 13 21
kk
k
.99305152024997656497... 1
1024
5
8
1
82 2 ( ) ( )
... ii eLieLi 2
22
2
.9944020304296468447... 6 2 2 26 2
22 3
21
log log( )
k
kkk 1
.99452678821883983884... 3
318 3 h J314
1 .9947114020071633897... 5
2
.99532226501895273416... erfk k
k k
k
22 1 2
2 1
2 1
0
( )
!( )
1 .9954559575001380004... dx
x x0
...
2
20
( log ) log sin
x xdx
x GR 4.421.1
... ( ) ( )
( )
( )
( )
91 45 5
187501
1
5 1
1
5 1
4
4 41
k k
k k k
10 .99557428756427633462... 7
2
21 .99557428756427633462... 7
211
22
2
1
k
k
kk
k
... 1
786432
3
8
5
8
1
8
7
84 4 4 4 ( ) ( ) ( ) ( )
= 11
4 1
1
4 15 51
( )
( )
( )
( )
k k
k k k
... 2
2 2
( )kk
k
... arctan k
kk 21
.99592331507778367120...
sinhsin ( ) sin ( )/ /
81 1 1
13
1 4 3 48
2 kk
7 .99601165442664514565... ( )kk
2
8
1 .99601964118442342116... 142 3844
3
0
ek
k kk
!( )
.99608116021237162307... 5207
49601160
8
8
J312
... 32
3
238
81 3 2
4
1
kk
kk
.99610656865... J310 g8
.99615782807708806401... 5
15365
1
2 1
5
50
( )( )
( )
k
k k AS 23.2.21, J345
.99623300185264789923... 127
12096008
127 8
128
18 1
81
( )( ) ( )k
k k J306
.. 219
4096 2
5
= 11
4 1
1
4 15 51
( )
( )
( )
( )
k k
k k k
... tanhtan
i
i
e e
e e
... log
/
( )
( ' )
/3
4 3 4 1
22 20
1 2
K k k
k k
dk GR 6.153
8 .99646829575509185269... 2 2 9 31
33
0
log ( )sin
cos
x x d
x
x
... ( )( )
12
2
2
kk
k
k k
... e e e e
e e
k
ekk
( )
( ), ,
4 3 2
6
5
1
26 66 26 1
1
15 0
... 3
8
11
22
7 3
4
16 6 1
2 2 1
2 2
32
log( )
( )
k k
k kk
=k k
kk
2
2
1
2
( ( ) )
... ( )kk
2
9
... 9 J312
.9980501956570772372... 3236
55801305 3
9
9
g J310
... 23
1296 31
1
6 1
1
6 1
4
4 41
( )
( )
( )
( )
k k
k k k
.99809429754160533077...
( )( ) ( )
8255 9
256
1 1
91
k
k k AS 23.2.19
.99813603811037497213... 1
2 11
2
22
tanh( )
e
eek k J944
.9983343660981139742... 33 3
1
3
128 2
1
64
1
8 5
1
8 3
( ) ( )k kk
17 .99851436155475931539... 1016
81
512
27
4
3
1 2 3
41
log( )( )( )k k k Hk
kk
... 3 2 11
220
logx
dx
... 19
2 3 5 31
1
4 1
1
4 1
2 4
14 61
( )
( )
( )
( )
k k
k k k 6 J327
... 9 2
... HypPFQk
k
k
, , , , , ,( )
( )!
1
6
1
3
1
2
2
3
5
6
1
46656
1
60
... ( )( )
( )( ) ( )6
1
491520
1
4
3
4
1
2 15 5
60
k
k k AS 23.2.21
.998777285931944... J314 h4
33 .99884377071514942382... 5 1
8
4
2
4
0
e k
k k
k
k
!( )
... 48983
4591650240
10
10
... 73
684288010
511 10
512
110 1
101
( )( ) ( )k
k k J306
... 6 2
.99915501340010950975...
4
4 42
1 1
11
4sin sinh
kk
4
... e e e
e
k
k
e e
k
2 1 2
1
1
2 1
/ sinh
( )!
... 7
11
1 .99935982878411178897... 3 2
120
7 4 3
216 1
log x dx
x
... 3 2
... 61
1843207
1
2 1
7
70
( )( )
( )
k
k k AS 23.2.21
... 2307 2
13107201
1
4 1
1
4 1
7
7 71
( )
( )
( )
( )
k k
k k k
... 11
1944 3
5
5
h J314
... 2305
933121
1
6 1
1
6 1
5
5 51
( )
( )
( )
( )
k k
k k k
.9997539139218932560...
sinhcosh sin ( ) sin ( )/ /
12
3
21 15
2 1 6 5 6
= 1112
2
kk
.99980158731305804717... ( )
( )!
1
70
k
k k
... 24611
165150720 21
1
4 1
1
4 1
8
81
( )
( )
( )
( )
k k
k k k 8 J327
.99984999024682965634... ( ) ( ) ( )81
330301440
1
4
3
47 7
=( )
( )
1
2 1 80
k
k k
... 1681
933120 31
1
6 1
1
6 1
6
61
( )
( )
( )
( )
k k
k k k 6 J327
... 6385
412876809
1
2 1
9
90
( )( )
( )
k
k k AS 23.2.21
... ( )
( )!
1
80
k
k k
... 301
524880 3
7
7
h J314
... 23337
1007769601
1
6 1
1
6 1
7
7 71
( )
( )
( )
( )
k k
k k k
... ( )
( )!
1
90
k
k k
...
188
8
)16(
)1(
)16(
)1(1
31410877440
25743
k
kk
kk
J329
.99999972442680776055... ( )
( )
1
100
k
k k !