beta gamma functions
DESCRIPTION
A lecture note on Beta and gamma functions..TRANSCRIPT
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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 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
![Page 3: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/3.jpg)
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
![Page 4: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/4.jpg)
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
![Page 5: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/5.jpg)
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
![Page 6: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/6.jpg)
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
![Page 7: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/7.jpg)
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
![Page 8: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/8.jpg)
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
![Page 9: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/9.jpg)
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
![Page 10: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/10.jpg)
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
![Page 11: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/11.jpg)
Exercise
(1)
∫ ∞−∞
e−k2x2dx
(2)
∫ ∞0
e−x3dx
(3)
∫ 1
0xm(log
1
x
)n
dx
N. B. Vyas Beta & Gamma functions
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Exercise
(1)
∫ ∞−∞
e−k2x2dx
(2)
∫ ∞0
e−x3dx
(3)
∫ 1
0xm(log
1
x
)n
dx
N. B. Vyas Beta & Gamma functions
![Page 13: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/13.jpg)
Exercise
(1)
∫ ∞−∞
e−k2x2dx
(2)
∫ ∞0
e−x3dx
(3)
∫ 1
0xm(log
1
x
)n
dx
N. B. Vyas Beta & Gamma functions
![Page 14: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/14.jpg)
Exercise
(1)
∫ ∞−∞
e−k2x2dx
(2)
∫ ∞0
e−x3dx
(3)
∫ 1
0xm(log
1
x
)n
dx
N. B. Vyas Beta & Gamma functions
![Page 15: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/15.jpg)
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
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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
![Page 17: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/17.jpg)
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
![Page 18: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/18.jpg)
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
![Page 19: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/19.jpg)
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
![Page 20: Beta gamma functions](https://reader036.vdocuments.mx/reader036/viewer/2022081719/557cbbd4d8b42a59078b45fd/html5/thumbnails/20.jpg)
Exercise
Ex. Prove that
∫ π2
0sinpθ cosqθ dθ =
1
2β
(p+ 1
2,q + 1
2
)
N. B. Vyas Beta & Gamma functions
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Exercise
Ex. Prove that
∫ ∞0
xm−1
(a+ bx)m+ndx =
β(m,n)
anbm
N. B. Vyas Beta & Gamma functions
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Relation between Beta and Gamma functions
β(m,n) =Γ(m)Γ(n)
Γ(m+ n)
N. B. Vyas Beta & Gamma functions
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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
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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