13. Gamma Function

1. Definitions, Properties

2. Digamma & Polygamma Functions

3. The Beta Function

4. Sterling’s Series

5. Riemann Zeta Function

6. Other Related Functions

Peculiarities:

1. Do not satisfy any differential equation with rational coefficients.

2. Not a hypergeometric nor a confluent hypergeometric function.

Common occurence:

In expansion coefficients.

13.1. Definitions, PropertiesDefinition, infinite limit (Euler) version :

1 2 3

lim1 2

z

n

nz n

z z z z n

0, 1, 2,z

1

limz n

nk

n k

z z k

1

1

1 lim1 1

z n

nk

n kz

z z k

1

1

lim1

nz

nk

n kn

n z k

1

limn

z

nk

kn

z k

z z

1

1 lim1 1

n

nk

n k

k

lim

1n

n

n

1

1

1

n

k

n n k

1 !n 1, 2, 3,n

1 1z z z

Definition: Definite Integral

Definition, definite integral (Euler) version :

1

0

t zz d t e t

Re 0z

2 2 1

0

2 s zd s e s

11

0

1ln

z

d ss

2 ; 2t s d t s d s

ln ; tt s e d t d s

2

0

12

2sd s e

1

2

, else singular at t = 0.

Equivalence of the Limit & Integral Definitions

Consider 1

0

, 1nn

ztF z n d t t

n

Re 0z

0

!lim 1 lim

! !

n kn

n nk

t n t

n n k k n

lim , ,n

F z n F z

0

lim!

kn

nk

t

k

te

z

1

1

0

, 1nz zF z n n du u u

tu

n

11 1

00

1 1z

n nz znu u n du u u

z

1

1

0

1nz zn

n du u uz

1

1

0

1 1

1 1z z nn n

n du uz z z n

1 1 1

1 1z n n

nz z z n z n

Definition: Infinite Product (Weierstrass Form)

Definition, Infinite Product (Weierstrass) version :

1

1lim ln

0.577 215 664 901

nm

nm

/

1

11z z n

n

zz e e

z n

Euler-Mascheroni constant

Proof : 1

limz n

nk

n kz

z z k

1

1lim 1

n

znk

z z

z n k

ln

1

lim 1n

z n

nk

zz e

k

ln /

1 1

1lim exp 1

n nz n z k

nm k

zz e z e

m k

/

1

1z z k

k

zz e e

k

Functional Relations 1 1z z z

1

0

t zz d t e t

Reflection formula :( about z = ½ )

1sin

z zz

Proof : 0 0

1 1 s z t zz z d s e s d t e t

Let u s t

sv

t

ds d t J du dv u v

u v

s sJ

t t

1

1

uvs

vu

tv

2

2

1 1

1

1 1

v u

v v

u

v v

21

u

v

2

0 0

1 11

zu v

z z du u e d vv

0 0

zs t s

d s d t et

20 1

zvd v

v

2 1

f (z) has pole of order m at z0 :

2

2 11

zz i

C

vd v e I

v

2

0

1 11

zvz z d v I

v

For z integers, set branch cut ( for v z ) = + x-axis :

22 Res ; 11

zvi v

v

1

2z

v

d vi

d v

12 i zi z e

1

22

1

i z

z i

eI i z

e

sin

z

z

1 1z z

0

1

0 01

1Res ;

1 !

mm

m

z z

df z z z f z

m d z

1

2z

3 1

2 2 2

21 1

2 2

1

2

Legendre’s Duplication Formula

211 2 2 1

2zz z z

1 1 3 3 1

2 2 2 2 2n n n

General proof in §13.3.

Proof for z = n = 1, 2, 3, …. :

1 1z z z

12 1 2 3 3 1

2nn n

1 !n n

12 1 !!

2nn

2 !!

2n

n 2

2 !!11 2 1 !!

2 2 n

nn n n

2

2 1

2 n

n

( Case z = 0 is proved by inspection. )

Analytic Properties

Weierstrass form : /

1

11z z n

n

zz e e

z n

has simple zeros at z

n,

no poles.

1

z (z) has simple poles at z

n,

no zeros.

0.46143 0.88560

changes sign at z n.

Minimum of for x > 0 is

Mathematica

Residues at z n

Residue at simple pole z n is

n z nR z n z

0lim n

z n

0

1lim

n

n

0

2lim

1

n

n n

1 1z z z

0

1lim

1 1n n

!

n

nRn

0

1lim

1 1n

n n

n + 1 times :

Schlaefli Integral

2 1 1

2 sin 1

t i

C

i

d t e t e

i e

Schlaefli integral :

Proof :

C1 is an open contour. ( e t for Re t . Branch-cut. )

0limt x

A AI d t e t d x e x

1

2

0limt x i

B BI d t e t d x e x e

2 1ie

2

11

0

ii etD D

I d t e t i d e

0

it e

211

0

ii d e

2 1

1 1

1

ie

0

0DI

if > 1

1

tA D BC

d t e t I I I 2 1 1iDe I

For Re < 1 , IA , IB , & ID are all singular.

However, remains finite.

( integrand regular everywhere on

C )

Factorial function :

(z) is the Gauss’ notation

1

2 1 1t iDC

d t e t e I

For Re > 1, ID = 0

reproduces the integral

represention.

1

2

11

1t

i Cd t e t

e

1

t t

C Cd t e t d t e t

0

1 lim xd x e x

! 1z z z

where

2

11

1t

i Cd t e t

e

is valid for all .

Example 13.1.1 Maxwell-Boltzmann Distribution

Classical statistics (for distinguishable particles) :

Probability of state of energy E being occupied is

/1 1E k T Ep E e eZ Z

Maxwell-Boltzmann distribution

Partition functionE

E

Z e 1E

p E

Average energy : E

E p E E 1 E

E

e EZ

1 Ed E g E e EZ

g(E) = density of states

Ideal gas : g E c E

1/2

0

EZ c d E E e

3/2 1/2

0

xc d x x e

x E 3/2 3 / 2c

3/2

0

EcE d E E e

Z

5/2 3/2

0

xcd x x e

Z

5/2 5 / 2

c

Z

1 5 / 2

3 / 2

3

2kT

gamma distribution

13.2. Digamma & Polygamma Functions

lnd

z zd z

Digamma function : 1 d z

z d z

1

limz n

nk

n kz

z z k

0

ln lim ln ln ! lnn

nk

z z n n z k

0

1lim ln

n

nk

z nz k

1 1

1 1

1 1 1lim ln

1

n n

nk k

nk z k k

1

1

1k

z

k z k

1

1

1lim ln

1

n

nk

nz k

1 0.57721566490153286060651209008240243104215933593992 50 digits

z = integer : 1

1 1

1k

nn k k

1

1

1n

k k

Mathematica

Polygamma Function

Polygamma Function : 1

1ln

m mm

m m

d z dz z

d z d z

1

11

11 !

mm

mk

mk

1, 2, 3,m

1

1

1m

m

mk

dz z k

d z

1

1 1

1k

zz k k

1 1

1

! 1m m

k

m z k

1! 1

mm m

= Reimann zeta function

Mathematica

Maclaurin Expansion of ln

1

1ln

mm

m

dz z

d z

1 1

1ln ln 1 ln

!

m m

mm z

z dz z

m d z

1

2

11 1 1

!

mm

m

zz

m

2

11

mm

m

zz m

m

11 ! 1

mm m m

1

Converges for

1 1z

Stirling’s series ( § 13.4 ) has a better convergence.

Series Summation

Example 13.2.1. Catalan’s Constant

Dirichlet series : ns

n

aS s

n

0 2 1

n

sn

sn

Catalan’s Constant : 2

0

2 0.915965594177219015052 1

n

n n

2 2

0 0

1 12

4 1 4 3m pm p

2

2 1

n m

n p

1

11

1!

1

mmm

k

z mz k

2 21 1

1 1 1 1 11

16 9 161 34 4

m pm p

1 11 5 1 1 71

16 4 9 16 4

0.91596559417721901505 20 digits

Mathematica

13.3. The Beta Function

,p q

B p qp q

Beta Function :

2 22 1 2 1

0 0

4 s p t qp q d s e s d t e t

2 2 1

0

2 s zz d s e s

cos

sin

s r

t r

r

r

s sds dt dr d

t t

cos sin

sin cos

rdr d

r

r dr d

2/2

2 1 2 1 2 1

0 0

4 cos sinp qr p qp q r d r d e r

/2

2 1 2 1

0

2 cos sinp qp q d

/2

2 1 2 1

0

, 2 cos sinp qB p q d

/2

2 1 2 1

0

! !2 cos sin

1 !m nm n

dm n

,B q p

1

1

p m

q n

Alternate Forms : Definite Integrals

/2

2 1 2 1

0

1, 1 2 cos sinp qB p q d

1

2 2 2

0

1, 1 cos cos sinp qB p q d 2cost 1

0

1qpdt t t

1

2 1 2

0

1, 1 2 1qpB p q d x x x 2t x

1/2

20

1 11, 1

1 11

p qu

B p q duu uu

1

ut

u

2

1

1 1

ud t du

u u

2

1

1du

u

1

11

tu

1/2

20 1

p

p q

udu

u

To be used in integral rep. of Bessel (Ex.14.1.17) & hypergeometric (Ex.18.5.12) functions

Derivation: Legendre Duplication Formula

1

0

1, 1 1qpB p q dt t t

1

1/21/2

0

1 1, 1

2 2zzB z z dt t t

2 1

1/2 1/2

1

11 1

2

zz z

d s s s

11

2t s

2 1

1/22

1

11

2

zz

d s s

2 11/22

0

12 1

2

zz

d s s

2

1 1 1,

2 2 2

z

B z

1

2 1 2

0

2 1qpd x x x

21 1 1 1

12 2 2 22 1 2 1

zz z z

z z

2

11 2 1

2 2 zz z z