This document discusses the Gamma and Beta functions. It defines them using improper definite integrals and notes they are special transcendental functions. The Gamma function was introduced by Euler and both functions have applications in areas like number theory and physics. The document provides properties of each function and examples of evaluating integrals using their definitions and relations.
Beta and gamma are the two most popular functions in mathematics. Gamma is a single variable function, whereas Beta is a two-variable function. The relation between beta and gamma function will help to solve many problems in physics and mathematics.
Beta and gamma are the two most popular functions in mathematics. Gamma is a single variable function, whereas Beta is a two-variable function. The relation between beta and gamma function will help to solve many problems in physics and mathematics.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
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Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
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Gradient , Directional Derivative , Divergence , Curl in mathematics
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Gradient , Directional Derivative , Divergence , Curl
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Maxwells equation and Electromagnetic WavesA K Mishra
These slide contains Scalar,Vector fields ,gradients,Divergence,and Curl,Gauss divergence theorem,Stoks theorem,Maxwell electromagnetic equations ,Pointing theorem,Depth of penetration (Skin depth) for graduate and Engineering students and teachers.
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
In this presentation we will learn Del operator, Gradient of scalar function , Directional Derivative, Divergence of vector function, Curl of a vector function and after that solved some example related to above.
Gradient in math
Directional derivative in math
Divergence in math
Curl in math
Gradient , Directional Derivative , Divergence , Curl in mathematics
Gradient , Directional Derivative , Divergence , Curl in math
Gradient , Directional Derivative , Divergence , Curl
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1. Beta & Gamma functions
Nirav B. Vyas
Department of Mathematics
Atmiya Institute of Technology and Science
Yogidham, Kalavad road
Rajkot - 360005 . Gujarat
N. B. Vyas Beta & Gamma functions
2. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
N. B. Vyas Beta & Gamma functions
3. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
These functions are very useful in many areas like asymptotic
series, Riemann-zeta function, number theory, etc. and also
have many applications in engineering and physics.
N. B. Vyas Beta & Gamma functions
4. Introduction
The Gamma function and Beta functions belong to the
category of the special transcendental functions and are
defined in terms of improper definite integrals.
These functions are very useful in many areas like asymptotic
series, Riemann-zeta function, number theory, etc. and also
have many applications in engineering and physics.
The Gamma function was first introduced by Swiss
mathematician Leonhard Euler(1707-1783).
N. B. Vyas Beta & Gamma functions
5. Gamma function
Definition:
Let n be any positive number. Then the definite integral
∞
e−x xn−1 dx is called gamma function of n which is
0
denoted by Γn and it is defined as
∞
Γ(n) = e−x xn−1 dx, n > 0
0
N. B. Vyas Beta & Gamma functions
6. Properties of Gamma function
(1) Γ(n + 1) = nΓn
N. B. Vyas Beta & Gamma functions
7. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
N. B. Vyas Beta & Gamma functions
8. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
N. B. Vyas Beta & Gamma functions
9. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
∞
Γn
(4) n = e−tx xn−1 dx
t 0
N. B. Vyas Beta & Gamma functions
10. Properties of Gamma function
(1) Γ(n + 1) = nΓn
(2) Γ(n + 1) = n!, where n is a positive integer
∞
2
(3) Γ(n) = 2 e−x x2n−1 dx
0
∞
Γn
(4) n = e−tx xn−1 dx
t 0
1 √
(5) Γ = π
2
N. B. Vyas Beta & Gamma functions
11. Exercise
∞
2 x2
(1) e−k dx
−∞
N. B. Vyas Beta & Gamma functions
13. Exercise
∞
2 x2
(1) e−k dx
−∞
∞
3
(2) e−x dx
0
1 n
1
(3) xm log dx
0 x
N. B. Vyas Beta & Gamma functions
14. Exercise
∞
2 x2
(1) e−k dx
−∞
∞
3
(2) e−x dx
0
1 n
1
(3) xm log dx
0 x
N. B. Vyas Beta & Gamma functions
15. Beta Function
Definition:
The Beta function denoted by β(m, n) or B(m, n) is defined as
1
B(m, n) = xm−1 (1 − x)n−1 dx, (m > 0, n > 0)
0
N. B. Vyas Beta & Gamma functions
16. Properties of Beta Function
(1) B(m, n) = B(n, m)
N. B. Vyas Beta & Gamma functions
17. Properties of Beta Function
(1) B(m, n) = B(n, m)
π
2
(2) B(m, n) = 2 sin2m−1 θ cos2n−1 θ dθ
0
N. B. Vyas Beta & Gamma functions
18. Properties of Beta Function
(1) B(m, n) = B(n, m)
π
2
(2) B(m, n) = 2 sin2m−1 θ cos2n−1 θ dθ
0
∞
xm−1
(3) B(m, n) = dx
0 (1 + x)m+n
N. B. Vyas Beta & Gamma functions