1. Unit-2 GAMMA, BETA FUNCTION
RAI UNIVERSITY, AHMEDABAD 1
Course: B.Tech- II
Subject: Engineering Mathematics II
Unit-2
RAI UNIVERSITY, AHMEDABAD

2. Unit-2 GAMMA, BETA FUNCTION
RAI UNIVERSITY, AHMEDABAD 2
Unit-II: GAMMA, BETA FUNCTION
Sr. No. Name of the Topic Page No.
1 Definition of Gamma function 2
2 Examples Based on Gamma Function 3
3 Beta function 5
4 Relation between Beta and Gamma Functions 5
5 Dirichlet’s Integral 9
6 Application to Area & Volume: Liouville’s
extension of dirichlet theorem
11
7 Reference Book 13

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GAMMA, BETA FUNCTION
 The Gamma function and Beta functions belong to the category of the
special transcendental functions and are defined in terms of improper
definite integrals.
1.1 Definition of Gamma function :
The gamma function is denoted and defined by the integral
Γ = ( > 0)
∞
1.2 Properties of Gamma function :
1) Γ( + 1) = Γ
2) Γ( + 1) = ! When m is a positive integer.
3) Γ( + ) = ( + − 1)( + − 2) … … … Γ , when n is a
positive integer.
4) Γ = 2 ∫ ( > 0)
∞
5)
Γ
= ∫ ( > 0)
∞
6) Γ = √
7) ∫ =
√∞
8) ∫ ( ) =
( )
( )
Γ( + 1)
2.1 Examples Based on Gamma Function:

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Example 1:Evaluate (− ).
Solution: We know that Γ( + 1) = Γ
 Γ − + 1 = − Γ −
 Γ = − Γ −
 √ = − Γ −
∴ − = − √ . __________Ans.
Example 2: Evaluate ∫ √ √∞
Solution: Let = ∫ √∞
__________(i)
Putting √ = ⟹ = so that = 2 in (i), we get
= ∫ ⁄
2
∞
= 2 ⁄
∞
= 2
∞
= 2Γ
= 2 ×
3
2
Γ
3
2
= 2 ×
3
2
×
1
2
Γ
1
2
=
3
2
√
∴ ∫ √ √∞
= √ ________Ans.

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Example 3: Evaluate ∫
∞
.
Solution: Let = ∫
∞
_______ (i)
Putting =
⟹ log =
⟹ =
1
log
⟹ = in (i), we have
= ∫
∞
= ( )
∫
∞
=
1
(log )
( )
∞
=
1
(log )
Γ( + 1)
∴ ∫
∞
= ( )
( + ) ________ Ans.
Example 4: Prove that ∫ ( ) =
!
Solution: We know that
∫ ( ) =
( )
( )
Γ( + 1) _______(i)
Now, ∫ ( ) = ∫ ( )
Putting = = 4 in (i), we get

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( ) =
(−1)
(4 + 1)
Γ(4 + 1)
=
Γ5
5
=
!
__________ proved.
2.2 EXERCISE:
1) Evaluate: (a) Γ − (b)Γ (c)Γ(0)
2) ∫
∞
3) ∫
4) ∫ ( )
3.1 BETA FUNCTION:
Definition: The Beta function denoted by ( , ) or ( , ) is defined as
( , ) = (1 − ) , ( > 0, > 0)
3.2 Properties of Beta function:
1) B(m,n) = B(n,m)
2) ( , ) = 2 ∫
3) ( , ) = ∫ ( )
∞
4) ( , ) = ∫ ( )
4.1 Relation between Beta and Gamma Functions:
Relation between Beta and gamma functions is
( , ) =
Γ .Γ
Γ( )

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 Using above relation we can derive following results:
 ∫ = , =
Γ .
Γ
 Γ = √
 Euler’s formula:
Γ . Γ(1 − ) =
sin
 Duplication formula:
Γ . Γ +
1
2
=
√ Γ(2 )
2
4.2 EXAMPLES:
Example 1: Evaluate ∫ − √
Solution: Let √ = ⟹ = so that = 2
1 − √ = ( ) (1 − ) (2 )
= 2 ∫ (1 − )
= 2 (10,6)
= 2
Γ Γ
Γ
= 2 ×
9! 5!
15!
=
× × × × ×
× × × × ×
=
× × ×
=
∴ ∫ 1 − √ = _________ Ans.

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Example 2: Find the value of .
Solution: We know that,
=
Γ .
2Γ
Putting = = 0, we get ∫ =
 [ ] ⁄
= Γ
 = Γ
 Γ =
 Γ = √ _______ Ans.
Example 3: show that ∫ √ =
Solution: We know that,
∫ =
Γ .
Γ
∫ √ = ∫
⁄
⁄
= ∫ ⁄ ⁄
On applying formula (1), we have
∫ √ =
Γ Γ
Γ
=
Γ Γ
Γ( )
= Γ Γ
∴ ∫ √ = Γ Γ __________ Ans.

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Example 4: Evaluate∫ ( + ) ( − )
Solution: Put = 2 cos 2 , then = −2 sin 2 in
(1 + ) (1 − )
= (1 + 2 ) (1 − 2 ) (−2 2 )
= (1 + 2 − 1) (1 − 1 + 2 ) (−4 )
= 4 2 . 2 .
= 2
∞
= 2
Γ Γ
2Γ
= 2
Γ( )Γ( )
Γ( )
__________Ans.
Example 5: Show that ( ) ( − ) = ( < < 1)
Solution: We know that
( , ) =
(1 + )
∞
Γ Γ
Γ( + )
=
(1 + )
∞
Putting + = 1 = 1 − , we get

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Γ(1 − ) Γ
Γ1
=
(1 + )
∞
Γ(1 − )Γ = ∫
∞
∵ ∫
∞
=
∴ Γ( )Γ(1 − ) = ______ proved.
4.3 EXERCISE:
1) Evaluate ∫ (1 − ) ⁄
2) Evaluate ∫ ( )
3) Evaluate ∫
4) Prove that Γ Γ = √2
5) Show that ( , ) = ( + 1, ) + ( , + 1)
5.1 DIRICHLET’S INTEGRAL:
If , , are all positive, then the triple integral
=
Γ(l)Γ(m)Γ(n)
Γ( + + + 1)
Where V is the region ≥ 0, ≥ 0, ≥ 0 and + + ≤ 1.
Note:
=
Γ(l)Γ(m)Γ(n)
Γ( + + + 1)
ℎ
Where V is the domain, ≥ 0, ≥ 0, ≥ 0 and + + ≤ ℎ
5.2 Corollary: Dirichlet’s theorem for n variables, the theorem status that

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… … …
=
Γ Γ Γ … Γ
Γ(1 + + + ⋯ + )
ℎ ⋯
Example 1: Prove that ∫
( )
( )
=
∞
Solution: Let = ∫
( )
( )
∞
 = ∫ ( )
∞
+ ∫ ( )
∞
 = + __________ (i)
Now, put = , when = 0, = 0; when = ∞, = 1
1 + = 1 + =
 =
( )
∴ =
1 −
. (1 − ) .
1
(1 − )
= ∫ (1 − )
= (5,10) _______(2)
And = ∫ . (1 − ) .
( )
= ∫ (1 − )
= (10,5) ________(3)
∴ = +
= (5,10) + (10,5) [Using(2) and (3)]
= (5,10) + (5,10) [ ( , ) = ( , )]

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= 2 (5,10)
=
Γ Γ
Γ
=
× !× !
!
=
× × × × × !
× × × × × !
= _______ Proved.
5.3 EXERCISE:
1) Find the value of ∫ ( )
2) Show that ∫
( )
( )
=
( , )
( )
3) ( + 1, ) = ( , )
6.1 Application to Area & Volume:
 Liouville’s extension of dirichlet theorem:
( + + )
=
Γ(l)Γ(m)Γ(n)
Γ(l + m + n)
( )
Example1: Show that ∭ ( )
= − , the integral being
taken throughout the volume bounded by
= , = , = , + + = .
Solution: By Liouville’s theorem when 0 < + + < 1

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( + + + 1)
=
( + + + 1)
(0 ≤ + + ≤ 1)
=
Γ Γ Γ
Γ( )
∫ ( )
u du
=
1
2 ( + 1)
= ∫ −
( )
+
( )
(Partial fractions)
= log( + 1) + −
( )
= 2 + 2 − 1 − −
= 2 −
∴ ∭ ( )
= − _______ Proved.
Example 2: Find the mass of an octant of the ellipsoid + + = ,
the density at any point being = .
Solution: Mass = ∭
= ( )
= ∭( )( )( ) _______ (1)
Putting = , = , = and + + = 1
So that = , = , =
Mass= ∭

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= ∭ , Where + + ≤ 1
= ∭
=
Γ Γ Γ
Γ
=
8 × 6
=
∴ = Ans.
6.2 EXERCISE:
1) Find the value of ∭ ( + + ) the integral extending
over all positive and zero values of , , subject to the condition
+ + < 1.

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2) Evaluate ∭ , integral being taken over all
positive values of , , such that + + ≤ 1.
3) Find the area and the mass contained m the first quadrant enclosed by
the curve + = 1 ℎ > 0, > 0 given that density at
any point ( ) is .
7.1 REFERENCE BOOK:
1) Introduction to Engineering Mathematics
By H. K. DASS. & Dr. RAMA VERMA
2) Higher Engineering Mathematics
By B.V.RAMANA
3) A text book of Engineering Mathematics
By N.P.BALI
4) www1.gantep.edu.tr/~olgar/C6.SP.pdf