A lecture note on Beta and gamma functions..

1. Beta & Gamma functions
Nirav B. Vyas
Department of Mathematics
Atmiya Institute of Technology and Science
Yogidham, Kalavad road
Rajkot - 360005 . Gujarat
N. B. Vyas Beta & Gamma functions

2. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
N. B. Vyas Beta & Gamma functions

3. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
These functions are very useful in many areas like asymptotic
series, Riemann-zeta function, number theory, etc. and also
have many applications in engineering and physics.
N. B. Vyas Beta & Gamma functions

4. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
These functions are very useful in many areas like asymptotic
series, Riemann-zeta function, number theory, etc. and also
have many applications in engineering and physics.
The Gamma function was first introduced by Swiss
mathematician Leonhard Euler(1707-1783).
N. B. Vyas Beta & Gamma functions

5. Gamma function
Definition:
Let n be any positive number. Then the definite integral
∞
e−x xn−1 dx is called gamma function of n which is
0
denoted by Γn and it is defined as
∞
Γ(n) = e−x xn−1 dx, n > 0
0
N. B. Vyas Beta & Gamma functions

6. Properties of Gamma function
(1) Γ(n + 1) = nΓn
N. B. Vyas Beta & Gamma functions

7. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
N. B. Vyas Beta & Gamma functions

8. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
N. B. Vyas Beta & Gamma functions

9. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
∞
Γn
(4) n = e−tx xn−1 dx
t 0
N. B. Vyas Beta & Gamma functions

10. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
∞
Γn
(4) n = e−tx xn−1 dx
t 0
1 √
(5) Γ = π
2
N. B. Vyas Beta & Gamma functions

11. Exercise
∞
2 x2
(1) e−k dx
−∞
N. B. Vyas Beta & Gamma functions

12. Exercise
∞
2 x2
(1) e−k dx
−∞
∞
3
(2) e−x dx
0
N. B. Vyas Beta & Gamma functions

13. Exercise
∞
2 x2
(1) e−k dx
−∞
∞
3
(2) e−x dx
0
1 n
1
(3) xm log dx
0 x
N. B. Vyas Beta & Gamma functions

14. Exercise
∞
2 x2
(1) e−k dx
−∞
∞
3
(2) e−x dx
0
1 n
1
(3) xm log dx
0 x
N. B. Vyas Beta & Gamma functions

15. Beta Function
Definition:
The Beta function denoted by β(m, n) or B(m, n) is defined as
1
B(m, n) = xm−1 (1 − x)n−1 dx, (m > 0, n > 0)
0
N. B. Vyas Beta & Gamma functions

16. Properties of Beta Function
(1) B(m, n) = B(n, m)
N. B. Vyas Beta & Gamma functions

17. Properties of Beta Function
(1) B(m, n) = B(n, m)
π
2
(2) B(m, n) = 2 sin2m−1 θ cos2n−1 θ dθ
0
N. B. Vyas Beta & Gamma functions

18. Properties of Beta Function
(1) B(m, n) = B(n, m)
π
2
(2) B(m, n) = 2 sin2m−1 θ cos2n−1 θ dθ
0
∞
xm−1
(3) B(m, n) = dx
0 (1 + x)m+n
N. B. Vyas Beta & Gamma functions

19. Properties of Beta Function
(1) B(m, n) = B(n, m)
π
2
(2) B(m, n) = 2 sin2m−1 θ cos2n−1 θ dθ
0
∞
xm−1
(3) B(m, n) = dx
0 (1 + x)m+n
1
xm−1 + xn−1
(4) B(m, n) = dx
0 (1 + x)m+n
N. B. Vyas Beta & Gamma functions

20. Exercise
π
2 1 p+1 q+1
Ex. Prove that sinp θ cosq θ dθ = β ,
0 2 2 2
N. B. Vyas Beta & Gamma functions

21. Exercise
∞
xm−1 β(m, n)
Ex. Prove that m+n
dx = n m
0 (a + bx) a b
N. B. Vyas Beta & Gamma functions

22. Relation between Beta and Gamma functions
Γ(m)Γ(n)
β(m, n) =
Γ(m + n)
N. B. Vyas Beta & Gamma functions

23. Exercise
π p+1 q+1
2
p q 1 Γ( 2 )Γ( 2 )
Ex. Prove that sin θ cos θ dθ =
0 2 Γ( p+q+2 )
2
N. B. Vyas Beta & Gamma functions

24. Exercise
π p+1 q+1
2
p q1 Γ( 2 )Γ( 2 )
Ex. Prove that sin θ cos θ dθ =
0 2 Γ( p+q+2 )
2
Ex. Prove that: B(m, n) = B(m, n + 1) + B(m + 1, n)
N. B. Vyas Beta & Gamma functions