GAMMA FUNCTIONS

Gamma Function:


The Gamma function is denoted by n , where n > 0 and it is defined as

n   e  x . x n1dx , n  0
0

Example: For n = 1
 
1   e . x dx   e  x . x 0 dx
x 11

0 0


1   e  x dx
0


e x 
1    e x 
1 0 0



1    e   e 0 

1    0  1

11
 
For n = 2 2   e . x dx   e  x . x dx
x 21

0 0

Integrating by parts
 
e x e x
2x  1 dx
1 0 0
 1




2

, 
x 
2    xe   e x
dx
0
0
x 
2    xe  1 ______  i    i  by first proof
0


x 
 Lim , 
x ex 
By L-Hospital rule
1 1 1
 Lim   0
x  e x e 

eq  i   2    0  0   1

 2 1

Some properties of Gamma function:
(i) n  1  n n  n! , n0

n  zx
(ii) nz e . x n1dx ; n, z  0
0
1 n1
 1
(iii) n    log  dy
0 y
 1
 yn
(iv) n  e dy
0

1
(v)  
2
Proof: (i) By definition of Gamma function

n   e  x . x n1dx
0

Replace n by n+1
 
x n11
n 1  e . x dx   e  x . x n dx
0 0


3