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Leonhard Euler's Integral: A Historical Profile of the Gamma Function: In Memoriam: MiltonAbramowitzAuthor(s): Philip J. DavisReviewed work(s):Source: The American Mathematical Monthly, Vol. 66, No. 10 (Dec., 1959), pp. 849-869Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2309786.Accessed: 17/08/2012 02:19
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19591
LEONHARD EULER'S INTEGRAL
849
FIG. 2: p=3,
q=1,
k=2a.
LEONHARD EULER'S
INTEGRAL:
A HISTORICAL PROFILE OF THE GAMMA FUNCTION
IN
MEMORIAM:
MITON ABRAMOWITZ
PHILIP J. DAVIS, National
Bureau of Standards,Washington, .
C.
Many people think that mathematical deas are static. They think
that the
ideas briginated t some time in the historicalpast and remain unchanged
for
all future imes. There are good reasonsfor uch a feeling.After ll, the
formula
for
he area
of
a circle was
7rr2
n
Euclid's day and at the present ime s
still rr2.
But to one who knows mathematics
from he inside, the subject has rather
the
feeling f a living thing. t growsdaily by the accretion of new information,t
changes daily by regarding tself
and the world fromnew vantage points, it
maintains a regulatorybalance by
consigning o the oblivion of irrelevancy
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850
LEONHARD
EULER
S INTEGRAL
[December
fraction
f its
past
accomplishments.
The purpose
of
this essay
is
to
illustrate
this process
of growth.We
select
one
mathematical
object,
thegamma
function,
nd show
howitgrew
n
concept
and
in content
from
he
time
of Euler
to the recent mathematical
treatise
of
Bourbaki, and how, in this growth, t partook of the general development of
mathematics
ver
the
past two and
a
quarter
centuries.
Of the so-called
"higher
mathematical
functions,"
he gamma
function
s
undoubtedly
the most
funda-
mental.
It is
simple
enough
for uniors
in college
to
meet
but deep enough
to
have
called
forth
ontributions
rom
he finestmathematicians.
And
it is
suffi-
ciently
compact
to
allow
its profile
o be sketched
within
the
space
of a
brief
essay.
The year
1729
saw
the
birth
ofthe
gamma
function
n a
correspondence
e-
tween a
Swiss
mathematician
n
St.
Petersburg
nd
a German
mathematician
in Moscow. The former: eonhardEuler (1707-1783), then22 yearsof age, but
to become
a prodigious
mathematician,
he
greatest
of the
18th century.
The
latter:
Christian
Goldbach
(1690-1764),
a savant,
a man ofmanytalents
and
in correspondence
withthe
leading
thinkers
f the
day.
As
a
mathematician
he
was
something
f
a
dilettante,
yet
he was a
man
who bequeathed
to the
future
a
problem
n
the theory f
numbers
o easy
to state
and
so
difficulto prove
that
even
to this
day
it
remains
on the
mathematical
horizon
as
a challenge.
The birthof
the
gamma
function
was
due
to
the
merging
f
several
mathe-
matical
streams.
The
first
was
that
of
interpolation
heory,
a very
practical
subject largelythe productof Englishmathematicians f the 17th centurybut
whichall
mathematicians
njoyed
dipping
into
from
ime
to
time.
The
second
stream
was
thatof
the
integral
alculus
and
of
the systematic
building
up
of
the
formulasof
indefinite
ntegration,
process
which
had
been going
on
steadily
for
many
years.
A
certain
ostensibly
impleproblem
of
interpolation
rose
and
was bandied
about
unsuccessfully
y
Goldbach
and
by
Daniel
Bernoulli 1700-
1784)
and even
earlier
byJames
Stirling
1692-1770).
The
problem
was
posed
to
Euler.
Euler
announced
his
solution
to Goldbach
in two
letters which
were
to
be the beginning
of an
extensive
correspondence
which
lasted
the duration
of
Goldbach's life.The firstetterdated October13,
1729
dealt with
the
interpola-
tion
problem,
while
the second
dated January
8,
1730 dealt with
integration
and tied the
two together.
Euler
wrote Goldbach
the
merest
utline,
but
within
a
year
he
published
all the
details
in
an
article
De
progressionibus
ranscendent-
ibus
seu
quarum
termini
enerales
lgebraice
dari
nequeunt.
his
article
can
now
be found
reprinted
n Volume
I14
of
Euler's
Opera
Omnia.
Since
the interpolation
problem
s the
easier
one,
let us
begin
with
it.
One
of the
simplest
sequences
of
integers
which
leads to an
interesting
heory
s
1,
1+2,
1+2+3,
1+2+3+4,
*
.
These
are the triangular
numbers,
o
called
because
they
represent
he
number
f
objects
which
can be
placed
in a
triangular
arrayof various sizes. Call the nthone T.. There is a formulaforT. whichis
learned
in
school algebra:
T.=
In(n+1).
What, precisely,
oes this formula
ccomplish?
n the first
lace,
it
simplifies
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1959]
LEONHARD
EULER S
INTEGRAL
851
computation
by reducing
large
numberof
additions to three
fixed
perations:
one of
addition, one of
multiplication,
nd
one of
division.Thus,
instead of
add-
ing the
first
undred ntegers o
obtain
T0oo0
e can
compute
Tloo=
(100)(100+1)
= 5050.
Secondly,
even
though
t doesn't
make
literal
sense to ask
for,
ay,
the
sum ofthefirst 1 integers, he formulaforT. producesan answerto this. For
whatever t is
worth, he
formula
yields
T51=
(5
)(5I+1)
=
17k.
In thisway,
the
formula
xtends the
scope of the
original
problem
to
values ofthe
variable
otherthan
those
forwhich it
was
originally
defined nd solves
the
problem of
interpolating
etween
the known
elementaryvalues.
This
type
ofquestion,
one which
asks
for n
extensionof
meaning,
cropped
up
frequently
n the
17th
and 18th
centuries.
Consider,for
nstance,
the
algebra
of
exponents.
The
quantity
am
is
defined
nitially
s the
product of
m
successive
a's.
This
definition as
meaningwhen
m is a
positive
nteger,
utwhat
would
a5l
be? The product of 5' successive a's? The mysteriousdefinitions '=1,
am/n
= /am,
a-"
=
l/am
which
solve
this
enigma
and
which are
employed
so
fruit-
fully n
algebra were
written
down
explicitlyfor the first
ime
by
Newton in
1676.
They are
justifiedby a
utility
whichderives
from
hefact that
the defini-
tion
leads to
continuous
exponential
functions
nd
that the law
of exponents
am
an
=am+n
becomes
meaningful orall
exponents
whether
positive
integers r
not.
Other
problems
of
this
type
proved harder.
Thus,
Leibnitz
introduced the
notation
dn
forthe
nth
iterateof the
operation
of
differentiation.
oreover,he
identified -l with
f
nd
d-n
with the iterated ntegral.Then he triedto breathe
some
sense into the
symboldn
when n is
any real value
whatever.
What, indeed,
is the
5'th
derivative of a function? his
question had to wait
almost
two cen-
turies
for
satisfactory
nswer.
THE
FACTORIALS
n:
1
2
3 4
5 6 7 8
n :
1
2
6
24
120 720
5040
40,320
...
FIG.
1
INTELLIGENCE
TEST
Question:
What number houldbe inserted
n
the ower ine
half
way
between
he
upper5
and
6?
Euler's
Answer:
87.8852
v.
Hadamard's
Answer:
80.3002
v
But
to
return
o
our
sequence
of
triangular
numbers. f we
change
the
plus
signs
to
multiplication
signs
we
obtain a new
sequence:
1,
1
* ,
1
* .3, * * *
This
is thesequenceoffactorials.The factorials re usuallyabbreviated 1 , 2 , 3 ,
.
and the first ive re
1, 2,
6, 24,
120.
They grow
n size
veryrapidly.
The
number
100
if
written ut
in
full would have 158
digits.
By contrast,
T1oo=5050
has a
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852
LEONHARD
EULER
S
INTEGRAL
[December
mere
fourdigits.
Factorials
are omnipresent
n
mathematics;
one can
hardly
open
a page
of mathematical nalysis
without
finding
t strewnwith them.
This
being
the case,
is it possible
to
obtain
an easy formula
for
computing
the
fac-
torials?
And
is it possible
to
interpolatebetween
the factorials?
What
should
5 be? (See Fig. 1.) This is the interpolationproblemwhich ed to the gamma
function,
he
interpolation
problem
of
Stirling,
f Bernoulli,
and of
Goldbach.
As we know,
these
two problems
are
related,
forwhen
one
has
a formula
here
is the possibility
of
inserting
ntermediate
values
into it. And now
comes
the
surprising
hing.There
is no, in fact
there
can be, no formula
forthe
factorials
which
s of the
simple
typefound
for
Tn.
This
is implicit n
the very
title
Euler
chose
for
his
article.
Translate
the
Latin
and
we have
On
transcendental
rogres-
sions
whose
eneral
erm
annot
be
expressed
lgebraically.
he solution
o
factorial
interpolation
ay
deeper
than
"mere
algebra."
Infinite
processes
were
required.
In orderto appreciatea littlebettertheproblemconfrontinguler it is use-
ful
to
skip
ahead
a bit
and
formulate
t
in an up-to-date
fashion:
find
reason-
ably simple
function
which
at the
integers
1,
2,
3,
-
*
-
takes
on
the factorial
values
1,
2, 6,
. -
a .
Now today,
a function
s a
relationship
etween
two sets
of
numbers
wherein
o
a number
of one
set is
assigned
a number
of the
second
set.
What
is
stressed
s
the relationship
nd not
the nature of
the rules
which erve
to
determine
he
relationship.
To help
students
visualize
the
function
oncept
in
its
full generality,
mathematics
nstructors
re
accustomed
to draw
a curve
full
of twists
and
discontinuities.
he more
of these
the
more
general
the
function
s
supposed to be. Given, then,the points (1,1), (2, 2), (3, 6), (4, 24), * * *and
adopting
the point
of
view
wherein
"function"
s what
we have
just
said,
the
problem
of
nterpolation
s
one
of
finding
curve
which
passes through
he
given
points.
This is
ridiculously
asy
to
solve.
It can
be done
in
an
unlimited
number
of ways.
Merely
take
a pencil
and
draw
some
curve-any
curve
will
do-which
passes
through
he points.
Such
a
curve
automatically
defines function
which
solves
the
interpolation
problem.
In this
way, too
free
an
attitude
as to what
constitutes
a function
solves
the problem
trivially
and
would enrich
mathe-
matics
but
little.
Euler's task
was
different.
n
theearly
18th
century,
function
was more
or
less
synonymous
with
a
formula,
nd
by
a formula
was meant
an
expression
whichcould
be
derived
from lementarymanipulationswith ddition,
subtraction,
multiplication,
division, powers,
roots,
exponentials,
ogarithms,
differentiation,
ntegration,
nfinite
eries,
.e.,
one whichcame from
he
ordinary
processes
of
mathematical
analysis.
Such
a
formula
was
called an
expressio
analytica,
an analytical
expression.
Euler's
task was to
find,
f he
could,
an
analytical
expression
arising
naturally
from
the
corpus
of mathematics
which
would
yield
factorials
when
a
positive
integer
was
inserted,
but which
would
still
be meaningful
or
othervalues
of
the variable.
It
is
difficult
o chronicle
the exact
course of scientific
iscovery.
This
is
particularlytrue in mathematicswhere one traditionallyomits fromarticles
and books
all accounts
of false
starts,
of
the
initial
years
of
bungling,
nd
where
one
may
develop
one's
topic
forward
r
backward
or
sideways
in order
o heighten
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854
LEONHARD EULER'S
INTEGRAL
[December
formulas
which occur in
the
originalpaper.
Euler wrote
a plain
f
for
ft.)
He
substituted /g for
and found
r1
g"+1 1-2
-n
(5)
I
xflx(1-x)ndx=
-
+
-
)dx
f +
(n + 1)g(
f +
g)(f + 2-g) * (f
+
n*g)
And so,
(6)*
1-2 - n
f
+
(n+)g
J
xflgdx(1-X)
n.
(f
+
g)(f
+
2-g) *
(f
+
n-g)
gf+1
He
observed that he
could
isolate the
1-2
...
n if he
set
f
=
1
and
g=O
in
the
left-hand
member,
ut that if he
did so, he would
obtain on
theright n
indeter-
minate form
which
he
writes
quaintjy as
rxll0dx(l-
x)n
(7)*
J
d
n+1
He
now
proceeded to
findthe
value of the
expression
7)
*. He
first
made the
substitutionxgl(f+g)
n
place
of x. This
gave
him
(8)*
9g
xfI(u+f)dx
f+g
in
place of
dx and hence, the
right-handmemberof
(6)
*
becomes
(9)*
f
n+(
f +g
dx(l
-
XgI(f+))n.
Once
again,
Euler
made
a
trial
setting
of
f
=
1,
g
=
0
having
presumably
re-
duced
this
integral
first o
(10)
~ ~~~f
(n
+
1)g
/
;
x
10)
(f +g)n+l
Jog(fg
dx,
and
this
yieldedthe indeterminate
(l )*
f
x
X0)n
He
now considered the related
expression
1 -xz)/z,
for
vanishing
z.
He
differ-
entiated
the numerator
nd
denominator,
s he
says, by
a known
l'Hospital's)
rule,and obtained
-
x-dzlx
(12)*
dz
(lx
=
log
x),
which for
z=0
produced
-lx.
Thus,
(13)* (1
-
x)/
=-
Ix
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1959]
LEONHARD EULER S INTEGRAL 855
and
(14)*
(1
-
X?)8On=
(-Ix)".
He thereforeoncluded that
(15)
nI
=
f
(-log x)ndx.
This gave himwhat
he wanted, an expression orn as an integralwherein alues
other than
positive
integersmay be substituted. The reader is encouraged to
formulate is own criticism f Euler's derivation.
Students in
advanced calculus generally meet Euler's integral first n the
form
00
(16)
r(x)
=
f
e-ttx-ldt, e
=
2.71828
-
This modification f
the integral 15) as well as the Greek
r
is due to Adrien
Marie Legendre
(1752-1833). Legendre calls the
integral 4) with which Euler
started his derivation
the firstEulerian integral and
(15) the second Eulerian
integral.The first
ulerian integral s currently nownas the Beta function nd
is now conventionallywritten
(17) B(m, n)
=
xm1(1 -x)'-ldx.
With the tools available
in
advanced
calculus,
it
is
readily
established
(how
easily the great achievements
of the
past
seem
to be
comprehended
nd
dupli-
cated ) that
the
integralpossesses meaning
when
x
>
0 and thus
yields
a certain
function
(x)
defined
forthese values.
Moreover,
(18) r(n + 1)
=
n
whenever
n
is a
positive nteger.*
t
is further stablished
that for ll
x>0
(19)
xr(x)
=
r(x + 1).
This is
the so-called recurrence
relation for the
gamma
function
and
in
the
years following
Euler
it
plays,
as
we
shall
see,
an
increasingly mportant
role
in
its
theory.
These
facts, plus perhaps
the
relationship
between
Euler's
two
types
of
integrals
(20) B(m, n)
=
r(m)r(n)/r(m
n)
and
the all
important
tirling
formula
*
Legendre's
otation hifts
he
rgument.
auss ntroduced
notation
r(x)
ree f
his efect.
Legendre's
otationwon
out,but
cQatinules
o
plague
many eople.
The
notations
, r,
and
can
all be found
oday.
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856
LEONHARD EULER'
S
INTEGRAL
[December
(21)
P(x)
e$xxl2V/(27r),
which gives
us a
relatively
simple approximate
expressionfor
r
x) when x is
large, are about all that
advanced calculus students
learn of the
gamma func-
tion.
Chronologically peaking,this puts
them at about
the year 1750. The play
has hardly begun.
Just s the simple
desire
to extend
factorials o values
in betweenthe ntegers
led to the
discoveryof the gamma function,
he desireto extend
it to negative
values
and to complex
values led to its further evelopment
and
to a morepro-
found interpretation.
Naive questioning, uninhibited
play with symbols may
have
been at
the
very
bottomof it. What is the value
of (-5g) ? What is the
value
of
V/(- 1)
?
In
the
early years
of the 19thcentury,
he action broadened
and
moved into the
complex plane (the
set
of
all
numbersof the formx+iy,
where
=
\/(
1))
and there t became
part
of the
general
development of the
theoryof functions f a complexvariable that was to formone of the major
chapters
n mathematics.
The move
to the
complex
plane
was initiated
by
Karl
Friedrich
Gauss
(1777-1855),
who
began
with
Euler's
product
as
his starting
point.
Many
famous names
are now involved and not
just
one
stage
of action
but
many
stages.
It would take too
long
to record
and
describe
each forward
step
taken. We
shall have to be contentwith
a broader
picture.
Three
important
facts
were
now
known:
Euler's
integral,
Euler's
product,
and
the
functional
or
recurrence elationshipxr(x)
=
r(x+ 1), x>0.
This
last
is
the generalization
of the obvious
arithmeticfact that for
positive
integers,
(n+ 1)n = (n+ 1) It is a particularlyusefulrelationship nasmuchas it enables
us by applying
t over and
over
again
to reduce
the
problem
of
evaluating
a fac-
torial
of
an
arbitrary
eal numberwhole
or otherwise o the
problem
of evaluat-
ing the
factorial
of
an
appropriate
number
ying
between
0 and
1.
Thus,
if
we
write
n
=
41
in the
above
formula
we
obtain
(41+
1)
=
5(42)
If we
could
only
find
ut
what
(4D)
is,
then
we
would
know that
(52)
is. This
process
of reduc-
tion
to
lower numbers
can
be
kept
up
and
yields
(22)
(5-)
=
(3/2)(5/2)(7/2)(9/2)(11/2)(1/2)
and sincewe have
(2)
=
41
/7rrom1) and (2), we can nowcomputeour answer.
Such
a device
is
obviously very
important
for
anyone
who
must do calcula-
tions
with the
gamma
function.
Other
information
s
forthcoming
rom the
recurrence
elationship.Though
the formula
n + 1)n
=
(n + 1)
as a
condensa-
tion of
the
arithmetic
identity (n+1)-1-2
. . .
n=1-2
- -
-
n-(n+1)
makes
sense
only
for
n
=
1, 2, etc.,
blind
insertions
f other
values
produce
interesting
things.
Thus, inserting
n
=
0,
we
obtain 0
= 1.
Inserting uccessively
n
=-2
n=-42
*
and
reducing pwards,
we discover
(23)
(-52)
=
(2/1)(-2/1)(-2/3)(-2/5)(-2/7)(-2/9)(1/2)
Since we already know
what (i) is, we can compute
(-5kL)
In
this way the
recurrence elationship
nables
us to
compute
the values of factorials
f
negative
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LEONHARD
EULER S INTEGRAL
857
numbers.
Turning
now to
Euler's integral,
t can be
shown that
forvalues
of the
vari-
able
less
than
0, the
usual theorems
fanalysis
do not
suffice o assign a
mean-
ingto the
integral,
for t is divergent.
On the
other
hand, it is meaningful
nd
yields a value if one substitutesfor x any complexnumber of the form +bi
where a>0. With
such substitutions
he
integraltherefore
yields a
complex-
valued
function
which s defined
or ll complex
numbers
n the right-half
fthe
complex plane and
which
coincideswith
the ordinary
gamma
function or
real
values.
Euler's
product
s
even stronger.
Withtheexception
of
0, -1, -2,
* *
any
complex
number
whatever
can be
insertedforthe variable
and
the infinite
product
will converge,yielding
value. And so
it appears that
we
have at our
disposal
a number
of
methods,conceptually
and operationally
differentor
ex-
tending
the domain of definition f the
gamma function.
Do these different
methodsyieldthe same result?They do. But why?
The answer
s
to be found
n the notion
of
an analytic function.
This
is
the
focal
point
of the
theory
of functions
f a complex
variable
and an outgrowth
of
the older notionof an analyticalexpression.
As we have hinted,
arlier
mathe-
matics
was
vague about
this notion,
meaning
by it a function
which arose
in a
natural
way
in mathematical nalysis.
When later it
was discovered
by J.
B. J.
Fourier (1768-1830)
that functions
fwide generality
nd functions
with
un-
pleasant
characteristics
ould be
produced
by the infinite
uperposition
f
ordi-
nary
sines
and cosines,
t became
clear
that the criterion
f "arising
n
a natural
way" would have to be dropped.The discovery imultaneouslyforced broad-
ening
of
the
dea of a function
nd a narrowing
f what
was
meantby an
analytic
function.
Analytic
functions re
not so
arbitrary n
their behavior.
On the
contrary,
they possess
strong internal
ties.
Defined
very precisely
as functions
which
possess
a
complex
derivative or
equivalently
as
functions
which possess
power
series
expansions
ao+a,(z-z0)+a2(z
-z0)2+
- -
-
they
exhibit
the
remarkable
phenomenon
of
"action at a distance."
This means
that
the
behavior of
an
analytic
function ver any
intervalno
matter
how small
is sufficient
o
deter-
minecompletely ts behavior everywhere lse; its potentialrangeof definition
and
its values
are
theoretically
btainablefrom
his
nformation. nalytic
func-
tions,
moreover,obey
the
principle
of the permanence
of
functional
relation-
ships;
if
an
analytic
function atisfies
n
some portions
of
its
region
of definition
a certain
functional
relationship,
hen
it must do
so
wherever
t is
defined.
Conversely, such a relationship
may be
employed
to
extend
its definition o
unkndwn
egions.
Our understanding
f
the process
of analytic
continuation,
s
this
phenomenon
is known, is based
upon the
work
of Bernhard
Riemann
(1826-1866)
and Karl
Weierstrass
1815-1897).
The complex-valued
function
which results
from
he
substitution f
complex
numbers
nto
Euler's
integral
s
an analytic function.The functionwhichemergesfromEuler's product is an
analytic
function.
The recurrence elationship
for the gamma
function
f satis-
fied n
some
region
must
be satisfied n
any other
region
to
which
the
function
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858
LEONHARD
EULER
S INTEGRAL
[December
can
be
"continued"
analytically
and indeed may
be
employed
to effect uch
ex-
tensions.
All portions
of the complex
plane,
with the
exception
of the
values
0,
-1, -2, * *
* are
accessible to
the complex
gamma
functionwhich has
be-
come
the
unique, analytic
extension
to complex
values
of Euler's
integral
see
Fig. 3).
THE GAMMA
FUNCTION
XX 14S 1
FIG.
2*
To understand
why
there should
be excluded
poilnts
observe that
r7(x)
-r'(x
+1)/Ix,
and as
x
approaches
0,
we obtainr
1(0)
=
1
/0.
This is
+
00or
-
0
depend'ing
whether
0 is
approached
through positive
or
negative
values.
The
*
From: H.
T.
Davis,
Tables
of the
Higher
Mathematical
unctions,
ol.
I, Bloomington,
Indi_ana,
933.
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LEONHARD
EULER'S
INTEGRAL
859
functional
equation (19)
then, induces
this
behavior over
and over
again at
each
of
the negative integers.
The (real) gamma
function
s comprised
of an
infinite
umberof disconnected
portionsopening
up and down alternately.
The
portions
corresponding
o negative values are each squeezed
into an
infinite
strip one
unit in width,
but the
major portion
whichcorresponds o positive x
and
which
contains
the
factorials
s of nfinite
idth see Fig. 2). Thus,
there
re
excluded
points for
the gamma
function t
which t
exhibitsfrom
he ordinary
(real
variable)
point of
view a somewhat
unpleasant
and
capriciousbehavior.
THE
ABSOLUTE
VALUE
OF THE COMPLEX
GAMMAFUNCTION,
EXHIBITING
THE
POLES
AT
THE NEGATIVE
INTEGERS
r
-7 -3
-2 Z
O X,
4
FIG.
3*
But from
he
complex point
of
view,
these
points
of
singular
behavior (singular
in the sense of
Sherlock
Holmes)
merit pecial
study
and
become an
important
part
of
the
story.
n
pictures
f
the
complex
gamma
function hey
show
up
as
an
infinite ow of
"stalagmites,"
each of
infinite eight the
ones
in
the
figure
re
truncated
out
of
necessity)
which become
more
and
more needlelike
as
they go
out to
infinitysee
Fig. 3).
They
are
known as
poles.
Poles
are
points
where
the
function
has an
infinite
ehavior
of especially
simple type,
a
behavior
which
is
akin to
that of
such
simple
functions
s
the
hyperbola
y
=
1/x
at
x
=
0
or of
y
=
tan x at x
=
r/2. The theoryof analytic functions s especially interested
*
From:E. Jahnke
nd
F. Emde,
Tafein
hoherer
unktionen,
th
ed.,
Leipzig,
948.
-
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860 LEONHARD EULER
S
INTEGRAL [December
in singular behavior, and devotes much space to
the
study
of the
singularities.
Analytic functions ossess many types of singularity
ut those with only poles
are
known as
meromorphic.
here
are also functionswhich are
lucky enough to
possess no singularities
for finite
rguments.
Such functionsform n
elite
and
are known as entire functions.They are akin to polynomialswhile the mero-
morphicfunctions re akin to the ratio of polynomials.
The
gamma function s
meromorphic. ts reciprocal, 1/r(x), has on the
contrary no excluded points.
There is no trouble anywhere.At the points 0, -1, -2, * * * it merelybecomes
zero. And the zero value which
occurs an
infinity
f
times,
s
strongly
eminis-
cent
of the
sine.
In the wake of
the
extension
to
the
complex many
remarkable identities
emerge, nd thoughsome
of them can
and were obtained without reference o
complex variables, they acquire
a
far
deeper
and richer
meaning
when
regarded
from he extended point of view. There is the reflection ormula f Euler
(24)
r(z)r(1
-
z)
=
lr/sinrz.
It
is
readily shown, using
the recurrence
elation
of
the
gamma function,
hat
the
product r(z)r(1 -z)
is a
periodic
function
f
period 2;
but
despite
the
fact
that
sin irz
s
one of
the
simplestperiodic
functions,
who could have
anticipated
the
relationship 24)? What,
after
all,
does
trigonometry
ave to do with the
sequence 1, 2, 6,
24 which started
the whole discussion?
Here is a fine
xample
of
the delicate
patterns
which
make the mathematics
of the
period
so
magical.
From the complex point of view, a partial reason for the identity ies in the
similarity
etween
zeros
of
the
sine and the
poles
of
the
gamma
function.
There is the
duplication
formula
(25) r(2z)
=
(2ir)-1I222112Fr(z)r(z
-1)
discovered
by Legendre
and extended
by
Gauss
in his
researches
on the
hyper-
geometric
function
o the
multiplication
ormula
(26) r(nz) =
(2ir)12(1-n)nnz-l12Pr(z)
+
)r
(z
+1)
.
r
(Z
f-
1)
There
are
pretty
formulas
forthe
derivatives of
the
gamma
function
uch as
1
1
1
(27)
d2
ogr(z)/dz2
-
+
+ +
z2
(Z
+1)
2
(z
+2)
2
This
is
an example
of
a
type
of infinite
eries out of
which G.
Mittag-Leffler
(1846-1927)
later created
his
theory
of
partial
fraction
developments
of
mero-
morphicfunctions. here is the intimaterelationshipbetween the gammafunc-
tion and the
zeta function
which has
been
of fundamental
mportance
n
study-
ing the
distribution f
the
prime numbers,
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LEONHARtD
EULER'S
INTEGRAL
861
(28)
t(z)
=
t(-
z)r(1
-
z)2irz-1
in
6z,
where
(29) ~(Z) + + +
2z 3z1
This
formula
has
some interesting
istory
related
to
it. It was
first roved
by
Riemann in
1859
and was
conventionally
ttributed
to him.
Yet in 1894
it
was
discovered
that a modified
ersion
of the
identity ppears
in somework
of
Euler
which
had been
done
in
1749.
Euler did
not
claim to
have
proved
the formula.
However,
he
"verified"
t for
ntegers,
or
,
and
for
3/2.
The
verification
or
2
is by direct
ubstitution,
ut for ll the othervalues,
Eulerworkswith
divergent
infinite
eries.
This
was
more than
100
years
in advance of
a firm heory
f such
series,but withunerringntuition,he proceeded to sum themby what is now
called
the method
of Abel summation.
The case 3/2
is even more
interesting.
There,
invoking
both divergent
eries and
numerical evaluation,
he
came
out
with
numerical
agreement
o
5
decimal
places
All this work convinced
himof
the truth
of
his
identity.Rigorous
modern
proofs
do
not
requirethe
theory
of
divergent
eries,
but the notions
of
analytic
continuation
re
crucial.
In
view
of
the
essential
unity
f
the
gamma
function
ver
the
whole
complex
plane
it
is
theoretically
nd aesthetically mportant
to have a formula
which
works
for ll complex
numbers.
One such formulawas supplied
in 1848
by F.
W.
Newman:
(30)
i/r(z)
=
ze't{
(1
+
z)e-z}
{
1
+
z/2)ez12}
*,
wherey
.57721
56649
.
X
This
formula
s
essentially
a
factorization
f
1/r(z)
and is much the same
as a
factorization
of
polynomials.
It exhibits
clearly
where the
functionvanishes.
Setting
each factorequal
to
zero
we find
that
1/r1(z)
s
zero for z
=0,
z
=
-1,
z
=
-2, *
.
-
In the hands
of
Weierstrass,
t
became the
startingpoint
of his
particular
discussion
of
the
gamma
function.
Weierstrass
was interested
n
how
functions ther
than
polynomials
may
be factored.
A
numberof solated factor-
izationswere thenknown.Newman's formula 30) and the older factorization
of
the
sine
(31)
sin
rz
=
rz(1
Z2)(1
- -
9 *
are
anaong
them.
The factorization
f
polynomials
s
largely
n
algebraic
matter
but
the
extension
to functions
uch
as
the
sine
which have an
infinity
f roots
required
the
systematic
building
up
of a
theory
of
infinite
roducts.
In
1876
Weierstrass
ucceeded
in
producing
n extensive
theory
of factorizationswhich
includedas special cases thesewell-knownnfinite roducts,as well as certain
doubly periodic
functions.
In addition
to
showing
the
roots of
1/P(z),
formula
30)
does much
more.
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LEONHARD
EULER
S INTEGRAL
[December
It shows
immediately
hat the reciprocal
of the gamma
function
s
a
much
ess
difficult
unction
o deal
with
than the
gamma
function
tself. t
is an
entire
function,
hat is, one
of
thosedistinguished
unctions
whichpossesses
no
singu-
larities
whatever
forfinite rguments.
Weierstrass
was
so
struck
by
the
advan-
tages to be gained by startingwith1/r(z) thathe introduced special notation
for
t.
He called
1/r(u+1)
thefactorielle
f u
and wrote
Fc(u).
The theory
f
functions
f
a
complex
variable
unifies hotch-potch
f
curves
and
a
patchwork
of
methods.
Within
this theory,
with its highly
developed
studies
of
infinite
eries of
various
types,
was
brought
to fruition
tirling's
un-
successful
attempts
at solving
the
interpolation
problem
for
the
factorials.
Stirling
had
done
considerable
work
with
infinite
eries
of the
form
A+Bz+Cz(z-1)+Dz(z-1)(z-2)+
*
This series is particularlyusefulforfitting olynomialsto values given at the
integers
=0,
1,
2,
*
- -
.
The
method
of
finding
he coefficients
,
B,
C,
. . .
was well
known.
But
whenan
infinitemount
of fitting
s
required,
much
more
than simple
formal
work
is
needed,
for we are
then
dealing
with
a bona
fide
infinite
eries whose
convergence
mustbe
investigated.
tarting
from he
series
1,
2,
6, 24,
.*. .Stirling
found
nterpolating
olynomials
via the
above
series.
The resultant
nfinite
eries
s divergent.
The
factorials
grow
too
rapidly
n
size.
Stirling
ealized
thisand
put
out the suggestion
hat
if
perhaps
one
started
with
the
logarithms
of the
factorials
nstead
of the
factorials
themselves
the
size
mightbe cut downsufficientlyorone to do something.There thematterrested
until
1900
when
Charles
Hermite
1822-1901)
wrote
down
the Stirling
eries
for
log
(i
+z):
z(z-1)
_____l)(z-2
(32)
log
r(
+
z)
=
1
log
2
+
1z2-3
(log3-21lg2)+
and showed that
this identity
s
valid whenever
z is
a
complex
number of
the
form
a
+ib
with a>0.
The identity
tself could
have
been written
down by
Stirling,
but
theproof
would
have
been another
matter.
An
even
simpler
tart-
ing point is the function 1(z) (d/dz) log r(z), now known
as
the
digamma
or
psi
function.
This
leads
to the
Stirling
eries
d
-
log r(z)
dz
(33)
_
(z
- 1) z
-2)
(z-
1)(z-
2)(z-
3)
(z
1)
2*
+
3.3
which
in 1847 was
proved
convergent
for
a>0
by
M.
A.
Stern,
a
teacher
of
Riemann.
All
these matters
re today
special
cases
of
the
extensive
heory
f the
convergenceof interpolation eries.
Functions
are
the
building
blocks
of mathematical nalysis.
In the
18th and
19th
centuries
mathematicians
devoted
much
time
and
loving
care to
develop-
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LEONHARD
EULER S INTEGRAL 863
ing
the properties and interrelationships
etween
special functions.
Powers,
roots, algebraic
functions, rigonometric
unctions, xponential
functions,
oga-
rithmicfunctions,
he
gamma function, he beta
function,
he
hypergeometric
function, he elliptic
functions, he theta function,
he
Bessel function, he
Matheiu function, he Weber function,Struve function, he Airy function,
Lame functions,
iterally
hundredsof special
functionswere singled
out for
scrutiny
nd their main featureswere drawn. This is an art which s not much
cultivated these days.
Times have
changed and emphasis
has shifted.Mathe-
maticians
on the wholeprefermore
abstract fare.
Large classes of functions
re
studied instead of individual
ones. Sociology
has replaced
biography.The field
of special functions,
s it is now known, s left
argely to
a small but ardent
group ofenthusiasts
plus those whose
work n physics or
engineering onfronts
them
directlywith
the necessity
of dealing with such matters.
The early 1950's saw the publicationof some very extensivecomputations
of
the
gamma
function
n the complex plane.
Led off n 1950by a six-place
table
computed
n
England,
it was
followed n Russia by the publication
of a
very ex-
tensive
six-place table. This in turn
was followed
n 1954 by the publication
by
the
National Bureau
of Standards
in Washingtonof a twelve-place
able. Other
publications
of the complex gamma function
nd related
functionshave ap-
peared in
this country, n England,
and in Japan. In the
past, the major
com-
putations of
the
gamma
function
had been confined to real
values.
Two fine
tables,
one
by Gauss
in
1813 and
one by Legendre n 1825,
seemed to answerthe
mathematicalneeds ofa century.Modern technologyhad also caught up with
the
gamma
function.The tables
of the 1800's-
were computed laboriously
by
hand,
and
the recent ones
by electronic
digitalcomputers.
But what touched off
his
spate
ofcomputational
activity?
Until
the
initial
labors of
H.
T. Davis of Indiana University
n the
early 1930's,
the
complex
values of
the
gamma
functionhad hardly been
touched. It was
one
of those
curious
turns of events wherein
the
complex
gamma
function ppeared
in the
solution
of
various theoreticalproblemsof atomic
and nuclear theory.
For
in-
stance,
the radial wave functions orpositive
energy
tates
in
a Coulomb
field
leads
to
a differential
quation
whose
solution
nvolves
the
complexgamma
func-
tion. The complex gamma function nters into formulasforthe scatteringof
charged
particles,
for
he
nuclearforces
etween
protons,
n Fermi's
approximate
formulafor the
probability
of
-radiation,
and
in
many
other
places.
The
im-
portance
of these
problems
o physicists
has had
the
side
effect
f
computational
mathematics
finally atchingup
with two and
a quarter centuries f theoretical
development.
As
analysis grew,
both creating pecial functionsnd
delineating
wide
classes
of
functions,
arious classificationswere
used
in
orderto
organize
them
for
pur-
poses
of
convenient tudy.
The earlier
mathematicians rganized
functions
rom
without,operationally, skingwhat operationsofarithmetic r calculus had to
be
performed
n
order
to
achieve
them.
Today,
there
s
a much
greater
endency
to
look
at functions rom
within, rganically,
onsidering
heir
construction
s
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LEONHARD
EULER S INTEGRAL
[December
achieved
and asking
what geometrical
haracteristics
hey possess.
In the
earlier
classification
we
have at the lowest
and
most
accessiblelevel, powers,
roots,
and
all that could
be
concocted
fromthem
by
ordinary algebraic
manipulation.
These came
to be known
as algebraic
functions.
he
calculus,
with ts
character-
istic operation of taking limits, ntroduced ogarithms nd exponentials,the
latter
encompassing,
as Euler
showed,
the
sines and cosines
of trigonometry
which had
been available
from
arlier periods of
discovery.
There
is an impas-
sable wall
between the
algebraic
functions
nd
the new
imit-derived
nes.
This
wall consists
n
the fact
that tryas
one
might o construct,
ay, a
trigonometric
function
out
of the finite
material of
algebra, one
cannot
succeed.
In more
technical anguage,
the
algebraic
functions
re
closed with
respect
to the
proc-
esses of algebra,
and
the trigonometric
unctions
re forever
beyond its
pale.
(By
way
of a simple
analogy:
the even integers re
closed
with respect
to
the
operationsofaddition,subtraction, nd multiplication;you cannot producean
odd integer
from
he set of even
integers
using these
tools.)
This led
to the con-
cept
of transcendental
functions.
These are functionswhich
are
not algebraic.
The transcendental
functions ount
among their
members,
he trigonometric
functions,
he
logarithms,
he
exponentials,
he elliptic
functions,
n short,
prac-
tically
all
the
special
functionswhich
had been singledout
for
pecial study.
But
such
an
indescriminate
dumping
produced
too large
a class
to handle.
The
transcendentals
had to be
split
further
or
onvenience.
A major
tool of
analysis
is the
differentialquation,
expressing
he relationship
between
a function
nd
its rate ofgrowth. t was found hat somefunctions,ay the trigonometricunc-
tions,
although
they
are transcendental
nd do not
therefore
atisfy
n algebraic
equation,
nonetheless
atisfy
differential quation
whosecoefficients
re alge-
braic.
The solutions
of
algebraic
differentialquations
are an
extensive
though
not all-encompassing
lass of
transcendental
unctions.
hey
countamong
their
members a good
many
of
the
special
functions
which
arise
in mathematical
physics.
Where
does
the
gamma
function it nto this? It
is not an
algebraic
function.
This
was
recognized
early.
It
is
a
transcendental
function.
But for long
while
it
was
an
open
question
whetherthe gamma
function
atisfied
an
algebraic
differential
quation.
The
question
was settlednegatively n 1887 by 0. Holder
(1859-1937).
It
does not. It
is
of
a
higherorder
of transcendency.
t
is
a
so-
called transcendentally
ranscendent
unction,
nreachable by
solving
algebraic
equations,
and
equally
unreachable
by
solving algebraic
differential
quations.
The
subject
has
interested
many people
through
he
years
and
in
1925
Alexander
Ostrowski,
now
Professor meritusof the
University
f
Basel,
Switzerland,gave
an
alternate
proof
of
H6lder's
theorem.
Problems
of classification re
extremely
difficult
o handle.
Consider, for
instance,
the
following:
Can the equation
X7
+ 8x + 1
be solved
with
radicals?
Is ir transcendental?Can fdx/X/(x31) be foundin terms ofspecified lemen-
tary
functions?
Can the
differential
quation
dy/dx (1/x)
+ (l/y)
be
resolved
with
quadratures?
The
general
problems
of which these
are
representatives
re
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18/22
1959]
LEONHARD
EULER'S
INTEGRAL
865
even today
far
from
solved and this
despite famous
theories such
as Galois
Theory,
Lie
theory, heoryof Abelian integrals
which have derived
from
uch
simple questions. Each individual problemmay be
a one-shot ffair o be solved
by individual methods nvolving ncredible ngenuity.
HADAMARD'S
FACTORIAL FUNCTION
6
5
4
3
FIG. 4
There
re
infini'telyany
unctions
hich
roduce
actorilals.
he function
F(x)
=
6/r(1
-
x)) (d/dx)
log {r((
-
x)/2)/r(
-
A
is
an ent'ire
nalytic
unctilonhich oincideswith he
gamma
functiont the
pos'itive
ntegers.
It satisfieshe functional
quationF(x
+1)
=
xF(x)
+(1
/rJi(
x)).
We
return
once
again
to our
interpolationproblem.
We have
shown
how,
stri'ctlypeaking, here re an unlimitednumber f solutions o thi's roblem.To
drive
this
point
home,
we
might
mention
a curious solution
given
in
1894
by
Jacques
Hadamard
(1865-
).
Hadamard
found
a
relativelysilmple
ormula
involving
the
gamma
functionwhich
also
produces
factorialvalues at
the
posi-
tive
integers. See F'igs.
1
and
4.)
But
Hadamard's function
1 d
(34)
iog=(lr-x)
(-
in
strong
contrast to
the gamma function
tself,
possesses no
singularitiies
ny-
where n the finite
omplexplane.
It is an entire
analytic
solution to the inter-
polation
problem
and
hence,
from the
function
theoretic
point
of
view,
is
a
simpler
olution. n view of all this
ambiguity,why
then should
Euler's
solution
-
8/10/2019 Davis 1959
19/22
866
LEONHARD
EULER'S
INTEGRAL
[December
be considered
the solution
par
excellence?
From the
point
of
view of integrals,
the
answer is
clear.
Euler's
integral
appears
everywhere
nd
is
inextricably
ound to a
host of
special
functions.
ts
frequency
nd simplicity
make it
fundamental.
When the
chips
are down,
it is
the veryform f the integraland of its modificationswhich end it utility nd
importance.
For the interpolatory
oint
of
view,
we can
make no such
claim.
We
must take
a
deeper
look at
the
gamma
function
nd show
that
of all the
solutions
of
the interpolation
roblem,
t,
in
some
sense,
s the simplest.This
is
partially
a
matterof mathematical
esthetics.
A PSEUDOGAMMA
UNCTION
82
-
_
- _
_-
o
-
8
.
_ _
6
-
- -
-
4
- -
_
0
2
4
6
FIG. 5
The function
llustrated
roduces
actorials,
atisfies he functionalquation
of
the
gamma
function,
nd
is convex.
We have already
observed
that
Euler's
integral
satisfies
the
fundamental
recurrence
quation,
xr
x)
=
r
(x+1),
and that
this
equation
enables
us
to com-
pute
all
the
real values
of the
gamma
function
from
knowledge
merely
of
its
values
in
the
intervalfrom
to
1. Since the
solutionto the
interpolation
roblem
is
not
determined
uniquely,
t
makes
sense
to
add to the
problem
more
condi-
tions and to
inquire
whether
he
augmented
problem
then
possesses
a
unique
solution. If it does, we will hope that the solutioncoincides withEuler's. The
recurrence
elationship
s
a
natural
condition
to add.
If
we do
so,
we
find hat
the
gamma
function
s
again
not
the
only
function
which atisfies
his
recurrence
-
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20/22
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21/22
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8/10/2019 Davis 1959
22/22
1959]
EQUATIONS
WITH CONSTANT COEFFICIENTS
869
Though
the
numericalvalue of
y
s knownto hundreds
of
decimal places,
it is
not
known at the time
of writing
whethery is or
is not a rational
number.
An-
other
problem
of this sort deals
with the values
of the gamma function
tself.
Though,
curiously nough,
the product
F(1/4)/1Yw
can be proved
to be trans-
cendental, t is not knownwhetherF(1/4) is even rational.
GeorgeGamow,
the
distinguishedphysicist,
uotes Laplace
as saying that
when
the known areas of
a
subject
expand, so
also do its frontiers.
aplace
evidently
had
in mind the
picture
of a circle
expanding
in
an
infinite lane.
Gamow
disputes
this for
physics
and has
in mind the
picture
of a circle expand-
ingon a spherical
urface.
As thecircle expands,
ts boundary
first xpands,but
later
contracts.
This writer
grees
with
Gamow as far as mathematics
s
con-
cerned.
Yet the record
s this:
each
generation
has
found
something
f
interest
to say
about
the
gamma
function.
Perhaps
the next
generation
will also.
The
writer ishes o thank
rofessor
.
Truesdell
orhis
helpful
omments
nd criticism
nd
Dr.
H.
E.
Salzer
for numberfvaluable
references.
References
1. E. Artin, inffuhrung
ndie
Theorie
er
Gammafunktion,
eipzig,
931.
2. N. Bourbaki,
16ments
e
Mathematique,
ook IV,
Ch.
VII,
La Fonction
Gamma,
aris,
1951.
3. H.
T. Davis,
Tables
of he
Higher
Mathematical
unctions,
ol. I,
Bloomington,
ndiana,
1933.
4.
L. Euler,Opera
mnia,
ol. 14,
eipzig-Berlin,
924.
S. P. H. Fuss,Ed., Correspondanceath6matiquet Physique e QuelquesC6lbbres e6-
mbtres
u
XVIIIieme
iecle,
ome
,
St.Petersbourg,
843:
6. G.
H. Hardy,
Divergent
eries,
Oxford,
949,Ch.
II.
7. F.
L6sch
nd
F. Schoblik,
ie Fakultat
nd
verwandte
unktionen,
eipzig,
951.
8.
N.
Nielsen,
Handbuch
er
Theorie
er
Gammafunktion,
eipzig,
906.
9. Table
of
the
Gamma
Function
or
Complex
Arguments,
ational
Bureau
of Standards,
Applied
Math.
Ser.
34,
Washington,
954. Introduction
y
Herbert
.
Saizer.)
10. E.
T. Whittaker
nd
G. N. Watson,
A Course
fModernAnalysis,
ambridge,
947,
Ch.
12.
LINEAR
DIFFERENTIAL
OR
DIFFERENCE EQUATIONS
WITH
CONSTANT
COEFFICIENTS
H. L. TURRITTIN,
Institute
f
Technology,
niversity
f
Minnesota
1.,
ntroduction.*
olutions
of
a
system
of linear
differential
r
difference
equations
with
real
constant
coefficientsi1,
such
as
n
(1)
dxd/dt
aijxj
and
xi(t
+ h)
=E
aixj(t),
jl1 ji-
*
This paper
was
prepared
n
part
while
working
nder
USAF
contract
o.
AF
33(038)22893