Beta gamma functions

Beta & Gamma functions

Nirav B. Vyas

Department of MathematicsAtmiya Institute of Technology and Science

Yogidham, Kalavad roadRajkot - 360005 . Gujarat

N. B. Vyas Beta & Gamma functions

Introduction

The Gamma function and Beta functions belong to thecategory of the special transcendental functions and aredefined in terms of improper definite integrals.

These functions are very useful in many areas like asymptoticseries, Riemann-zeta function, number theory, etc. and alsohave many applications in engineering and physics.

The Gamma function was first introduced by Swissmathematician Leonhard Euler(1707-1783).


Introduction





Introduction





Gamma function

Definition:

Let n be any positive number. Then the definite integral∫ ∞0

e−xxn−1dx is called gamma function of n which is

denoted by Γn and it is defined as

Γ(n) =

∫ ∞0

e−xxn−1dx, n > 0


Properties of Gamma function

(1) Γ(n+ 1) = nΓn

(2) Γ(n+ 1) = n!, where n is a positive integer

(3) Γ(n) = 2

∫ ∞0

e−x2x2n−1dx

(4)Γn

tn=

∫ ∞0

e−txxn−1dx

(5) Γ

(1

2

)=√π



(1) Γ(n+ 1) = nΓn


(3) Γ(n) = 2

∫ ∞0

e−x2x2n−1dx

(4)Γn

tn=

∫ ∞0

e−txxn−1dx

(5) Γ

(1

2

)=√π



(1) Γ(n+ 1) = nΓn


(3) Γ(n) = 2

∫ ∞0

e−x2x2n−1dx

(4)Γn

tn=

∫ ∞0

e−txxn−1dx

(5) Γ

(1

2

)=√π



(1) Γ(n+ 1) = nΓn


(3) Γ(n) = 2

∫ ∞0

e−x2x2n−1dx

(4)Γn

tn=

∫ ∞0

e−txxn−1dx

(5) Γ

(1

2

)=√π



(1) Γ(n+ 1) = nΓn


(3) Γ(n) = 2

∫ ∞0

e−x2x2n−1dx

(4)Γn

tn=

∫ ∞0

e−txxn−1dx

(5) Γ

(1

2

)=√π


Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx


Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx


Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx


Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx


Beta Function

Definition:

The Beta function denoted by β(m,n) or B(m,n) is defined as

B(m,n) =

∫ 1

0xm−1(1− x)n−1dx, (m > 0, n > 0)


Properties of Beta Function

(1) B(m,n) = B(n,m)

(2) B(m,n) = 2

∫ π2

0sin2m−1θ cos2n−1θ dθ

(3) B(m,n) =

∫ ∞0

xm−1

(1 + x)m+ndx

(4) B(m,n) =

∫ 1

0

xm−1 + xn−1

(1 + x)m+ndx



(1) B(m,n) = B(n,m)

(2) B(m,n) = 2

∫ π2


(3) B(m,n) =

∫ ∞0

xm−1

(1 + x)m+ndx

(4) B(m,n) =

∫ 1

0

xm−1 + xn−1

(1 + x)m+ndx



(1) B(m,n) = B(n,m)

(2) B(m,n) = 2

∫ π2


(3) B(m,n) =

∫ ∞0

xm−1

(1 + x)m+ndx

(4) B(m,n) =

∫ 1

0

xm−1 + xn−1

(1 + x)m+ndx



(1) B(m,n) = B(n,m)

(2) B(m,n) = 2

∫ π2


(3) B(m,n) =

∫ ∞0

xm−1

(1 + x)m+ndx

(4) B(m,n) =

∫ 1

0

xm−1 + xn−1

(1 + x)m+ndx


Exercise

Ex. Prove that

∫ π2

0sinpθ cosqθ dθ =

1

2β

(p+ 1

2,q + 1

2

)


Exercise

Ex. Prove that

∫ ∞0

xm−1

(a+ bx)m+ndx =

β(m,n)

anbm


Relation between Beta and Gamma functions

β(m,n) =Γ(m)Γ(n)

Γ(m+ n)


Exercise

Ex. Prove that

∫ π2

0

sinpθ cosqθ dθ =1

2

Γ(p+12 )Γ(q+1

2 )

Γ(p+q+22 )

Ex. Prove that: B(m,n) = B(m,n+ 1) +B(m+ 1, n)


Exercise

Ex. Prove that

∫ π2

0

sinpθ cosqθ dθ =1

2

Γ(p+12 )Γ(q+1

2 )

Γ(p+q+22 )

Ex. Prove that: B(m,n) = B(m,n+ 1) +B(m+ 1, n)


Education

Beta gamma functions