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A lecture note on Beta and gamma functions..
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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
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