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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

8/10/2019 Davis 1959

3/22

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

8/10/2019 Davis 1959

4/22

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

8/10/2019 Davis 1959

5/22

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

8/10/2019 Davis 1959

6/22

8/10/2019 Davis 1959

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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

8/10/2019 Davis 1959

8/22

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.

8/10/2019 Davis 1959

9/22

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

8/10/2019 Davis 1959

10/22

1959]

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

8/10/2019 Davis 1959

11/22

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.

8/10/2019 Davis 1959

12/22

1959]

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.

8/10/2019 Davis 1959

13/22

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,

8/10/2019 Davis 1959

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1959]

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.

8/10/2019 Davis 1959

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862

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-

8/10/2019 Davis 1959

16/22

1959]

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

8/10/2019 Davis 1959

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864

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

8/10/2019 Davis 1959

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

8/10/2019 Davis 1959

20/22

8/10/2019 Davis 1959

21/22

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