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).

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).

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).

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx

N. B. Vyas Beta & Gamma functions

Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx

N. B. Vyas Beta & Gamma functions

Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx

N. B. Vyas Beta & Gamma functions

Exercise

(1)

∫ ∞−∞

e−k2x2dx

(2)

∫ ∞0

e−x3dx

(3)

∫ 1

0xm(log

1

x

)n

dx

N. B. Vyas Beta & Gamma functions

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)

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

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

N. B. Vyas Beta & Gamma functions

Exercise

Ex. Prove that

∫ π2

0sinpθ cosqθ dθ =

1

2β

(p+ 1

2,q + 1

2

)

N. B. Vyas Beta & Gamma functions

Exercise

Ex. Prove that

∫ ∞0

xm−1

(a+ bx)m+ndx =

β(m,n)

anbm

N. B. Vyas Beta & Gamma functions

Relation between Beta and Gamma functions

β(m,n) =Γ(m)Γ(n)

Γ(m+ n)

N. B. Vyas Beta & Gamma functions

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)

N. B. Vyas Beta & Gamma functions

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)

N. B. Vyas Beta & Gamma functions