Beta gamma functions

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
dened in terms of improper denite 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 rst introduced by Swiss
mathematicianLeonhard Euler(1707-1783).
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
dened in terms of improper denite 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 rst introduced by Swiss
mathematicianLeonhard Euler(1707-1783).
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
dened in terms of improper denite 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 rst introduced by Swiss
mathematicianLeonhard Euler(1707-1783).
N. B. Vyas Beta & Gamma functions

5. Gamma function
Denition:
Letnbe any positive number. Then the denite integral
Z
1
0
e
x
x
n1
dxis called gamma function ofnwhich is
denoted by nand it is dened as
(n) =
Z
1
0
e
x
x
n1
dx; n >0
N. B. Vyas Beta & Gamma functions

6. Properties of Gamma function
(1) n+ 1) =nn
(2) n+ 1) =n!, wherenis a positive integer
(3) n) = 2
Z
1
0
e
x
2
x
2n1
dx(4)
n
t
n
=
Z
1
0
e
tx
x
n1
dx(5)

1
2

=
p

N. B. Vyas Beta & Gamma functions

7. Properties of Gamma function
(1) n+ 1) =nn
(2) n+ 1) =n!, wherenis a positive integer
(3) n) = 2
Z
1
0
e
x
2
x
2n1
dx(4)
n
t
n
=
Z
1
0
e
tx
x
n1
dx(5)

1
2

=
p

N. B. Vyas Beta & Gamma functions

8. Properties of Gamma function
(1) n+ 1) =nn
(2) n+ 1) =n!, wherenis a positive integer
(3) n) = 2
Z
1
0
e
x
2
x
2n1
dx(4)
n
t
n
=
Z
1
0
e
tx
x
n1
dx(5)

1
2

=
p

N. B. Vyas Beta & Gamma functions

9. Properties of Gamma function
(1) n+ 1) =nn
(2) n+ 1) =n!, wherenis a positive integer
(3) n) = 2
Z
1
0
e
x
2
x
2n1
dx(4)
n
t
n
=
Z
1
0
e
tx
x
n1
dx(5)

1
2

=
p

N. B. Vyas Beta & Gamma functions

10. Properties of Gamma function
(1) n+ 1) =nn
(2) n+ 1) =n!, wherenis a positive integer
(3) n) = 2
Z
1
0
e
x
2
x
2n1
dx(4)
n
t
n
=
Z
1
0
e
tx
x
n1
dx(5)

1
2

=
p

N. B. Vyas Beta & Gamma functions

11. Exercise
(1)
Z
1
1
e
k
2
x
2
dx
(2)
Z
1
0
e
x
3
dx(3)
Z
1
0
x
m

log
1
x

n
dx
N. B. Vyas Beta & Gamma functions

12. Exercise
(1)
Z
1
1
e
k
2
x
2
dx
(2)
Z
1
0
e
x
3
dx(3)
Z
1
0
x
m

log
1
x

n
dx
N. B. Vyas Beta & Gamma functions

13. Exercise
(1)
Z
1
1
e
k
2
x
2
dx
(2)
Z
1
0
e
x
3
dx(3)
Z
1
0
x
m

log
1
x

n
dx
N. B. Vyas Beta & Gamma functions

14. Exercise
(1)
Z
1
1
e
k
2
x
2
dx
(2)
Z
1
0
e
x
3
dx(3)
Z
1
0
x
m

log
1
x

n
dx
N. B. Vyas Beta & Gamma functions

15. Beta Function
Denition:
The Beta function denoted by(m; n) orB(m; n) is dened as
B(m; n) =
Z
1
0
x
m1
(1x)
n1
dx;(m >0; n >0)
N. B. Vyas Beta & Gamma functions

16. Properties of Beta Function
(1)B(m; n) =B(n; m)
(2)B(m; n) = 2
Z
2
0
sin
2m1
cos
2n1
d
(3)B(m; n) =
Z
1
0
x
m1
(1 +x)
m+n
dx(4)B(m; n) =
Z
1
0
x
m1
+x
n1
(1 +x)
m+n
dx
N. B. Vyas Beta & Gamma functions

17. Properties of Beta Function
(1)B(m; n) =B(n; m)
(2)B(m; n) = 2
Z
2
0
sin
2m1
cos
2n1
d
(3)B(m; n) =
Z
1
0
x
m1
(1 +x)
m+n
dx(4)B(m; n) =
Z
1
0
x
m1
+x
n1
(1 +x)
m+n
dx
N. B. Vyas Beta & Gamma functions

18. Properties of Beta Function
(1)B(m; n) =B(n; m)
(2)B(m; n) = 2
Z
2
0
sin
2m1
cos
2n1
d
(3)B(m; n) =
Z
1
0
x
m1
(1 +x)
m+n
dx(4)B(m; n) =
Z
1
0
x
m1
+x
n1
(1 +x)
m+n
dx
N. B. Vyas Beta & Gamma functions

19. Properties of Beta Function
(1)B(m; n) =B(n; m)
(2)B(m; n) = 2
Z
2
0
sin
2m1
cos
2n1
d
(3)B(m; n) =
Z
1
0
x
m1
(1 +x)
m+n
dx(4)B(m; n) =
Z
1
0
x
m1
+x
n1
(1 +x)
m+n
dx
N. B. Vyas Beta & Gamma functions

20. Exercise
Ex.
Z
2
0
sin
p
cos
q
d=
1
2


p+ 1
2
;
q+ 1
2

N. B. Vyas Beta & Gamma functions

21. Exercise
Ex.
Z
1
0
x
m1
(a+bx)
m+n
dx=
(m; n)
a
n
b
m
N. B. Vyas Beta & Gamma functions

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

23. Exercise
Ex.
Z
2
0
sin
p
cos
q
d=
1
2
(
p+1
2
)(
q+1
2
)
(
p+q+2
2
)
Ex. B(m;n) =B(m;n+ 1) +B(m+ 1;n)
N. B. Vyas Beta & Gamma functions

24. Exercise
Ex.
Z
2
0
sin
p
cos
q
d=
1
2
(
p+1
2
)(
q+1
2
)
(
p+q+2
2
)
Ex. B(m;n) =B(m;n+ 1) +B(m+ 1;n)
N. B. Vyas Beta & Gamma functions