Proc. London Math. Soc.-1929-Watson-293-308

8/7/2019 Proc. London Math. Soc.-1929-Watson-293-308

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APPROXIMATIONS CONNECTED WITH


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294 G. N. WATSON [May lOr

function of a continuous positive variable x. When this is done, | and

J are still the upper and lower bounds for y; but. for a certain rangeof values of x, namely 0 < x


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1928.] APPROXIMATIONS CONNECTED WITH ex. 295

3. We shall now study the properties of u and U, qua functionsof t. It is easy to see that

udU _ U du_ w_ d?U _ U dhidt U-V dt~ 1-u' df ~ {(J-1)3' ~dF ~ (1-uf

It is possible to express u, U and t as explicit functions of a para-meter v by writing U = uc?'\ and then

u = v(cothr 1), U = y

sinh vt = v coth v 1 + l o g

but the complexity of the values of t and dt/dv in terms of v makes-this representation comparatively useless for practical purposes.

It may also be remarked that, as a consequence of Legendre'sdeduction* from Biirmann's theorem,

6 - T i + 2l + si

(which is certainly valid when 0 ^ u


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29 6 G. N. WATSON [May 10,

while, if z is now regarded as a complex variable, L7 is one of the branches

of w the many-valued function ofz defined by the equation

w l \ogto = \z*.

The only branch points of w, qua function of z, are points at whichdw/dz is zero or infinite, and, at all such points,w = 1 so that z? = 4nni(n = l, 2, 3, ...)

Hence the expansions of U and u are convergent when

The only possible singularities of w on the circle of convergence| . = 2-V/T are consequently

now a simple computation shows that

is not equal to 1 when z = '2c ** i^/ir, so that, in point of fact, if zapproaches the circle of convergence from inside, the points 2e5lr/N/

!7 arethe only singularities* of the branch U.

Near z = 2e3ff


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1928.] APPROXIMATIONS CONNECTED WITH e' :. 297

when n is large; this coefficient is equal to

1 .3 . 5 . . . (271-3 ) co8 j (n+ l )7 r ^ ,_)n c o 8 ^ ( n + l ) y{ ' 2 . 4 . 6 . . . ( 2 / t ) 2w-^7ri(M-1> ~( } 2 ' t - M ' l n 0 i '

by Stir ling 's theorem . Hence th e coefficient of t*" in the expansion ofUis approximately equal to

4. Now that the expansions ofu and U in ascending powers of t have'been obtained, it is possible to investigate the asymptotic expansion ofy.On integration by parts we have

The last integral is convergent whenx ^ 0, whereas previous integrals.'have been convergent only when x>().

Now d?(U-\-u)/dt2 is bounded when t ^ 0; for, when t is small, it isexpansible in a series of positive integral powers oft, convergent when0 ^ fc


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298 G. N. WATSON [May 10,

It is evident from this result that k is greater than 2/21 for all suffi-

ciently large values of x.

5. In order to obtain definite bounds for y and k, as opposed to-approximations, we need inequalities satisfied by d?(U-\-u)ldtf.

An inequality which is effective in dealing with y is

df + df < u '

but, before proving it, we shall establish the more elementary inequality

since the method required to prove it indicates the method which isappropriate for the former inequality.

(I) From the formulae of 3 it is evident that the inequality

+>is true if u+[7 2[7 ^ 0, i.e. if^ U

To prove that this is the case, define v by the equation

U

2*7-1

and then we have to prove that u ^ v.It is evident that

The function on the right vanishes when [7 = 1; and, when U > l rits derivate, which is 8([7 1) 2/(2[J I) 2, is positive. Hence the function,itself is positive when [ 7 > 1 , so that

Since ue x~ u is an increasing function of u, it follows from this inequality that u^-v, and therefore we have proved that

dt + dt


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1928] APPROXIMATIONS CONNECTED WITH e*. 29i>

(II) The inequality involving second differential coefficientsis rathermore troublesome; it is evidently adequate to prove that

U u

and for this purpose we shall prove that the function

U u

(which vanishes when t = 0) has a positive differential coefficient with,respect to t when t > 0.

It is easy to prove that

^ T U u 1 __ U ( 1 3 | , u f 1 , 8 )dt L g (u-if~ l 8 a-iif] ~ u-i \u~ u-i'i + l-u i u + 1 ^ >

= (U u)(


8/16


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1928.] APPROXIMATIONS CONNECTED WITH e* . 801

found preferable to use the expansion of $() in ascending powers of t.

This expansion to a large number of terms was obtained by taking theformulae of 3 :

2vcothi>, t = v c o t h t > l + l o g - ,

expanding the expressions on the right in powers of 3, reverting thesecond expansion so as to get v2 in series of powers of t and substitutingin the first expansion. The procedure, which involved considerablelabour, seemed to be the simplest method of determining the series

which, after two differentiations, gave the series for


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3 0 2 G. N . WATS ON [May 10,

But the differential coefficient with respect toU of the function on the

left has the negative value

2C7+4 (TT , . m

and so the function itself is negative when L7 exceeds the value forwh ich the function is zero. It is found by trial that

217+1 C72 4C7 rT" /T T , , . . 185TT i + tTT n a

l o 8 US l o g ( ? 7 l ) + l o g - ^ -

vanishes when U is about 11'8 and t is about 8'4. Hence it is nega tivewhen t > 8'4, and so

1tf-1-lostr 8I 185^g

is a decreasing function of t when t > 8"4.Hence t~ l log 0(i) is expressible as the sum of two functions which

are both decreasing when t ;> 8 -4 , and therefore t"1 log 8"4.

When 0 ^ t < 8 ' 4 , t he values of rU o g ^ f t ) shown in the Tab leindicate that the function varies continuously witht, the first and seconddifferences being reasonably steady throughout the range of the Table;and there is no singularity of this function in the complex domain oftnear this range of values of t to cause any abrupt variation of the func-tion between successive entries. A singularity which could cause thefunction to have a maximum concealed between successive entries

without producing any visible effect on the differences of the first fourorders would be a singularity of a kind which could not be expected tooccur in a function of so simple a type of function as the logarithmic-exponential type of which t"1 log (t) is a specimen.

It will therefore be assumed that the only maxim um of f~l log (t)in the interval 0 ^ t ^8*4 is the maximum near t = 7'2. whose presenceis evident from the Table.

With this assumption, the equation

log (t) _2_t 21

has only one positive root (the root t = 0 being, of course, excepted).This root is beyond the range of the.Table; its value is about 563, andit will be called a.


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1928.] APPROXIMATIONS CONNECTED WITH e*. 808

AI i.i i- log 6() 8

Also the equation ^ j1 1

- = jz

has only two positive roots b and c say (b 0

Then % ( ) + ^ ^ = e-*'(a-t)Mt) dt > 0,

since the integrand is never negative.Consequently eaxxi(x) ^ s a n increasing function of x ; and therefore,

since its value when x = 0 is - **, it is always positive.Hence, on the hypothesis that fi(t) = 0 has only one positive root, it

has been proved that k ^ 2/21.

Next write 0)-exp (~ ^ *) =

so that

Then

* 0,

since th e integrand is never negative.

That is to say

and so


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304 G. N. WATSON [May 10;

is an increasing function of x; it is easy to see tha t it tends to ao a&

a;->0, and it tends to -\- as z - > o o ; it therefore has one (and onlyone) zero, say x x0.

Then

that is to say

d r

Hence, as x increases from 0 to oo, e^xafa) decreases to a m inimumat Xoand then increases up to + o o . Since \3;(0) = 0, this minimum value-of e^xaC^) *s negative, and therefore eftrxa() n a s o n e ( a n (i only o n e )zero, say x = xx, other than x = 0 ; and xx ^> x0.

It is now evident that

XateXO (0


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1028.] APPROXIMATIONS CONNECTED WITH r 1 '. 305

8. The modification of the original integral ( 2) for y which is appro-

priate when x is small is

y = l-\-r.\ e 'duJ

Jo 2.r

Tn this integral write

6 ~ 1 + + +2! + m! +

where 0


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G. N. WATSON [May 10,

l ' J R /

Solutions of ne ] -" =Ue l-l' = e- 1 , with values of


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1928.] APPROXIMATIONS CONNECTED WITH e*. 307

Solutions of tie1"" = Uel~u c~', with values of


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30 8 APPROXIM ATIONS CONNECTED WITH e'\

Solutions of tie 1 - = Uel-V = e- \ with values of (t) = 1 |2 ( U H - _ \8 \(U I) 3 (1 w) 3J

t

6 0

6 16-26-36-46-5

6 66-76-8G-97 0

7-17-2737-4757-67-77-87-98 0

8-18-2838 48-58-68-78-88 99 0

9 19 29-39 49 59 69 79 89 9

10.0

10110210310-4105106107108109110

n

000091271

0000825790000747140O006760000006116a0-00055339

0 000500700-00045303000040J90

0-00087088000033558

000030363000027473000024858000022492000020361000018414000016661000015076000013641000012343

0-000111680000101050-000091430000082730000074860000067730-000061290-000055450-00005018000004540

000004108000vJ037170000033630000030430000027540-000024920000022550000020400000013460 10001670

000001511000001367000001237009001120000001013000000917000000829000000750000000679000000614

U

92215423

9-33362339-44554309-55730599-66891629-7803781

9-891695610 0028723101139119

10-224817910-3355936

10-4462423

10-556767110-667170910-777456610-887627010-997684811 107632511-217472611-327207611-4368397

11-546371211-655804311-765141011-874383411-983533512-092593112-201564212-810448412-419247512-5279632

12636597112745150912 853625912962023813 070346013-178593813-286768713-394871913-502904813-6108686

13-718764613-826593813-934357614-042056914-149692914-257266714-364779314-472231814579625014-6869600

02645747

0-25817030-2519625024594540-24011340-2344606

0-22898170-22367110-2185235021353350-2086962

0-2040065

0-199459501950505019077490-1866281018260580-1787038017491810-17124450-1676793

0-16421880-16085940-1575975

. 0-1544299015135330-14836460-14546070-14263880-13989610-1372297

0-134637201321161012966380-12727810-12495670-12269750-1204983011835720-11627230-1142416

0- L122635011033610-10845780 10662710-1048423010310200-10140480-09974920-09813410-0965581

-*-' log,

Proc. London Math. Soc.-1929-Watson-293-308

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