The Gamma Function

ECE 6382

Notes are from D. R. Wilton, Dept. of ECE

1

David R. Jackson

Fall 2021

Notes 14

y

( )zΓ

0z =

x

The Gamma Function

2

The Gamma function appears in many expressions, including Bessel functions, etc.

It generalizes the factorial function n! to non-integer values and even complex values.

It appears in the method of steepest descent (a method for obtaining the asymptotic expansion of a class of integrals).

Definition 1

3

1 2 3( ) lim , 0, 1, 2,( 1)( 2) ( )n

znz n zz z z z n→∞

⋅ ⋅Γ ≡ ≠ − −

+ + +

Definition # 1

This definition gives the Gamma function a nice property, as proven on the next slide:

( )( ) 1 !n nΓ = −

Definition 1 (cont.)

1

1 2 3( ) lim , 0, 1, 2,( 1)( 2) ( )

1 2 3( 1) lim ( ) lim( 1)( 2) ( 1) ( 1)

( 1) ( )

1 2 3(1) lim

n

n n

n

z

z

nz n zz z z z n

n nzz n zz z z n z n

z z z

n

→∞

→∞ →∞

→∞

+

⋅ ⋅Γ ≡ ≠ − −

+ + +⋅ ⋅

Γ + = = Γ+ + + + + +

Γ + = Γ

⋅ ⋅Γ =

⇒

Note that 1 2 3 n⋅ ⋅

1 (2) 1 (1) 1,( 1)

(3) 2 (2) 2 1, (4) 3 (3) 3 2 1, (5) 4 (4) 4 3 2 1, .

nn

= Γ = ⋅Γ =+

Γ = ⋅Γ = ⋅ Γ = ⋅Γ = ⋅ ⋅ Γ = ⋅Γ = ⋅ ⋅ ⋅ etc

, and

4

Factorial property:

( )( ) 1 !n nΓ = − ( 1) !n nΓ + =orHence

Definition 2

5

This is the Euler-integral form of the definition.

Leonard Euler

Note:Definition 1 is the analytic continuation of definition 2 from the right-half plane

into the entire complex plane (except at zero and the negative integers).

( ) ( )ln ln1 1 1 1

1 1 = 00

iyiy t iy tz x x x

z x t

t t t t e t e

t t x

− − − −

− −

= = =

⇒ = ⇒ > for the integral to converge at

Note:

0

1( ) , Re 0t zz e t dt z∞

− −Γ ≡ >∫

Definition # 2

Equivalent Integral Forms

( )

0

0

1

0

2

1

2 1 2

1

( ) , Re 0

( ) 2 , Re 0

1( ) ln , Re 0 ln 1 /

t z

s z

z

z e t dt z

z e s ds z t s

z ds z t ss

∞

∞

− −

− −

−

Γ ≡ >

Γ = > =

Γ = > =

∫

∫

∫

The following three integral definitions are all equivalent :

(let )

(let )

6

Equivalence of Definitions 1 and 2

( )

( ) ( ) 1

2

0

1 1

0 0

1 1

1

lim 1

( , ) 1 ; ( , )

1 2 3 1( , ) 1

( 1)( 2) ( 1)

n

n

nt

nnz t z

nz z z z n

z n

ten

tF z n t dt F z n e t dtn

tw nn

n nF z n n w w dw n w dw

z z z z n

z

→∞

→∞

−

∞− − −

− + −

+

≡ −

≡ − → =

=

⋅ ⋅ −= − =

Γ

+ + + −

∫ ∫

∫

Use ,

Define

Letting and integrating by parts t , imes

1

1

0

) ( )lim ( ,n

F n zz→∞

= Γ

∫

Factor appearing in Definition #1

Hence

7

Equivalence of definitions #1 and #2

(Please see next slide.)

8

( ) ( )

( )

( )

1

0

11

01

1

01

1

0

1 1

0 1

1

z zn n

zn

n z

w wI w n w dwz z

wn w dwz

n w w dwz

−

−

−

= − − − −

= + −

= −

∫

∫

∫

( )

11

01

dvu

dw

n zI w w dw−≡ −∫

Integration by parts development:

Integrate by parts once:

Equivalence of Definitions 1 and 2 (cont.)

( )( ) ( )1

0

11 2 3 2 11

( 1)( 2) ( 1)n n z nn n n

I w w dwz z z z n

− + −− − ⋅ ⋅= −

+ + + − ∫

9

( )

( ) ( ) ( ) ( )

( )( ) ( )

( )( ) ( )

1

0

11

011 11 2

01

2 1

01

2 1

0

1

1 1 1 11 1

10 1

1

11

1

n z

z zn n

n z

n z

nI w w dwz

n w n ww n w dwz z z z

n nw w dw

z z

n nw w dw

z z

−

+ +− −

− +

− +

= −

= − − − − −+ +

−= + −

+

−= −

+

∫

∫

∫

∫

After n times:

( )

11

01

dvu

dw

n zI w w dw−≡ −∫

Integrate by parts twice:

Equivalence of Definitions 1 and 2 (cont.)

Definition 3

1

1 1( )

zz n

n

zze ez n

γ∞ −

=

= + Γ ∏

10

Definition # 3

The Weierstrass product form can be shown to be equivalent to definitions #1 and #2.

0.5772156619γ = where is the Euler -Mascheroni constant.

Euler Reflection Formula

( ) (1 )sin

z zz

ππ

Γ Γ − =

11

Note: We can use this along with definition #2 to find Γ(z) for Re(z) < 0.

x

y1 z−z

1 / 2x =

Geometric interpretation of reflection formula:The two points are reflections about the x = 1/2 line.

Euler Reflection Formula

1( ) , Re 0, 0, 1, 2,sin (1 )

z z zz z

ππ

Γ = < ≠ − −Γ −

1/ 2 x− ( )1 1/ 2x− −

12

( ) (1 )sin

z zz

ππ

Γ Γ − =

Set z = 1/2:

(1 / 2) πΓ =

A special result that occurs frequently is Γ(1/2).

To calculate this, use the reflection formula:

Euler Reflection Formula (cont.)

Summary of Factorial Properties

13

( ) ( ) ( ) ( ) ( )! 1 2 3 2 1n n n n= − −

( )0

! 1 t xx x e t dt∞

−= Γ + = ∫

( )0

! 1 t zz z e t dt∞

−= Γ + = ∫

Integers

Real numbers

Complex numbers

1x > −

( )Re 1z > −

Summary of Factorial Generalization

14

Summary of Factorial Generalization (cont.)

Complex numbers1, 2z ≠ − −

( ) ( )0

! 1 Re 1

1( )sin (1 )

t zz z e t dt z

zz z

ππ

∞−= Γ + = > −

Γ =Γ −

∫

+

Summary of Factorial Properties (cont.)

Pole Behavior

15

Simple poles are at n = 0, -1, -2, -3,…

Use

( 1)( 1) ( ) ( ) zz z z zz

Γ +Γ + = Γ ⇒ Γ =

( )

( )

( 2)( 2) 1 ( 1) ( 1)1

( 2)( )1

( 2)( )1

zz z z zz

zz zzzz

z z

Γ +Γ + = + Γ + ⇒ Γ + =

+Γ +

⇒ Γ =+

Γ +⇒Γ =

+

Γ(z) has simple pole at z = 0Residue = 1

Γ(z) has simple pole at z = -1Residue = -1

( 1) ( ) & (1) 1z z zΓ + = Γ Γ =Recall:

Pole Behavior (cont.)

16

( )

( ) ( ) ( )

( ) ( ) ( )

( 4)( 4) 3 ( 3) ( 3)3

( 4)2 1 ( )3

( 4)( )1 2 3

zz z z zz

zz z z zz

zzz z z z

Γ +Γ + = + Γ + ⇒ Γ + =

+Γ +

⇒ + + Γ =+

Γ +⇒Γ =

+ + +

Γ(z) has simple pole at z = -2Residue = +1/2

( )

( ) ( )

( ) ( )

( 3)( 3) 2 ( 2) ( 2)2( 3)1 ( )

2( 3)( )1 2

zz z z zz

zz z zz

zzz z z

Γ +Γ + = + Γ + ⇒ Γ + =

+Γ +

⇒ + Γ =+

Γ +⇒Γ =

+ +

Γ(z) has simple pole at z = -3Residue = -1/6

17

Residues at Poles

( ) ( )1Res

!

n

nn−

Γ − =

Hence

Pole Behavior (cont.)

( ) ( ) ( ) ( )( 1)( )

1 2 3z nz

z z z z z nΓ + +

Γ =+ + + +

In general (after n steps), we will have:Γ(z) has simple pole at z = -n

( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( )

( 1)Res Lim

1 2 3

11 2 3 1

11 3 2 1

11 3 2 1

z n

z n

n

n nn z n

z z z z z n

z z z z z n

n n

n n

→−

=−

Γ − + +Γ − = + + + + +

=+ + + + −

=− − + − − −

−=

−

Plot of Gamma Function

18Note: There are simple poles at z = 0, -1, -2,…

y

( )zΓ

0z =

x

( ) ( )1Res

!

n

nn−

Γ − =

Plot of Gamma Function (cont.)

19

Γ(x) and 1 / Γ(x)

In fact, 1 / Γ(z) is analytic everywhere.

Note: Γ(x) never goes to zero.

Asymptotic Form of Gamma Function

20

Sterling’s formula (asymptotic series for large argument):

( ) 2 3 41 1 1 139 5712 1

12 288 51840 2488320z z

w

z z ez z z zz

π −

Γ + + − − +

( ), argz z→∞ = constant

Valid for

( ) 3 51 1 1 1ln ln ln2 2 12 360 1260

zz z z zz z zπ

Γ − − + − + +

Taking the ln of both sides, we also have

( )2 3

ln 12 3

w ww w+ = − + −Note :