dr. nirav vyas betagamma functions.pdf

8/20/2019 Dr. Nirav Vyas betagamma functions.pdf

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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


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Introduction

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



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Introduction


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



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Introduction


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 rst introduced by Swiss

mathematician Leonhard Euler

(1707-1783).



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Properties of Gamma function

(1) Γ(n + 1) = nΓn



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



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(3) Γ(n ) = 2

∞

0e − x

2x 2n − 1dx



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(3) Γ(n ) = 2

∞

0e − x

2x 2n − 1dx

(4) Γn

t n =

∞

0e − tx x n − 1dx



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(3) Γ(n ) = 2

∞

0e − x

2x 2n − 1dx

(4) Γn

t n =

∞

0e − tx x n − 1dx

(5) Γ12

= √ π



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Exercise

(1) ∞

−∞

e − k2 x 2 dx


i


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Exercise

(1) ∞

−∞

e − k2 x 2 dx

(2) ∞

0 e− x 3

dx


E i


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Exercise

(1) ∞

−∞

e − k2 x 2 dx

(2) ∞

0 e− x 3

dx

(3) 1

0x m log

1x

n

dx


E i


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Exercise

(1) ∞

−∞

e − k2 x 2 dx

(2) ∞

0 e− x 3

dx

(3) 1

0x m log

1x

n

dx


B t F ti


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Beta Function

Denition:The Beta function denoted by β (m, n ) or B (m, n ) is dened as

B (m, n ) = 1

0x m − 1(1 − x )

n − 1dx, (m > 0, n > 0)


Properties of Beta F nction


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Properties of Beta Function

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




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

(2) B (m, n ) = 2 π

2

0sin 2m − 1θ cos 2n − 1θ dθ




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

(2) B (m, n ) = 2 π

2

0sin 2m − 1θ cos 2n − 1θ dθ

(3) B (m, n ) = ∞

0

x m − 1

(1 + x)m + ndx



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Exercise


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Exercise

Ex. Prove that

π

2

0sin pθ cos qθ dθ =

1

2β

p + 1

2 ,

q + 1

2


Exercise


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Exercise

Ex. Prove that

∞

0

x m − 1

(a + bx)m + ndx =

β (m, n )

a n bm


Relation between Beta and Gamma functions


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Relation between Beta and Gamma functions

β (m, n

) =

Γ(m )Γ( n )

Γ(m + n )


Exercise


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Exercise

Ex. Prove that

π

2

0

sin pθ cos q θ dθ = 1

2

Γ( p+12 )Γ(q +1

2 )

Γ( p+ q +2

2 )


Exercise


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Exercise

Ex. Prove that

π

2

0

sin pθ cos q θ dθ = 1

2

Γ( p+12 )Γ(q +1

2 )

Γ( p+ q +2

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


dr. nirav vyas betagamma functions.pdf

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