This document discusses the gamma and beta functions. It defines the gamma function and lists some of its key properties. Examples are provided to demonstrate how to evaluate integrals using gamma function properties. The beta function is then defined and its relationship to the gamma function explained. Dirichlet's integral theorem and its extension to multiple dimensions is covered. Applications to finding volumes and masses are demonstrated. References for further reading on gamma and beta functions are listed at the end.

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