BETA FUNCTION

Beta function:
The Beta function denoted by (m, n  where m,n > 0 and is defined as

1
n1
  m, n    x m1.1  x  dx
0



Some properties of Beta function:
(i)   m, n     n, m  (symmetric property)


 
x n1 x m1
(ii)   m, n    m n
dx   n m
dx
0 1  x  0 1  x 



2
2 m 1 2 n 1
(iii)   m, n   2   sin    cos  d
0




10

, a a
Important calculus property   f  x  dx   f  a  x  dx
0 0


Proof: (i) L.H .S    m, n 
1
n 1
  x m1.1  x  dx
0

1 a a
m1 n 1
  1  x  .1  1  x   dx   f  x  dx   f  a  x  dx
0 0 0

1
m1 n1
  1  x  .1  1  x  dx
0

1
m1
  x n1 1  x  dx
0


  m, n     n, m 
Proof: (ii) By the definition of Beta function
1
n 1
  m, n    x m1.1  x  dx ___  i 
0
1 1
Let x   dx  2
dy
1 y 1  y 
1
1 y 
x
y   as x  0
y  0 as x  1
Put all values in (i)
0 m1 n 1
 1   1  1
  m, n      .1   . 2
dy

1 y   1 y  1  y 

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