This document discusses the beta and gamma functions. It introduces that the beta and gamma functions were introduced by Leonhard Euler in the 18th century to generalize the factorial function to non-integer and complex values. It defines the beta function using an improper definite integral and lists some of its applications in asymptotic series, the Riemann zeta function, and number theory. The document then evaluates the beta and gamma functions and lists some of their properties and the relationship between the two functions.

1. E.M.G. YADAVA WOMEN’S COLLEGE, MADURAI – 625 014.
(An Autonomous Institution – Affiliated to Madurai Kamaraj University)
Re-accredited (3rd Cycle) with Grade A+ and CGPA 3.51 by NAAC
Mrs.S.Manimozhi,
Assistant professor of physics,
Department of Physics,
E.M.G.Yadava women’s college,Madurai-14
Beta Function
and
Gamma Function

2.  It is introduced by Swiss mathematician
Leohard Euler(1707-1783)
 It simplifies the factorial function n to non intger
values and even complex values
 Gamma and Beta are special type of
transcendental functions
 It is improper definite integral
 Where it is applied in areas of
 Asymptotic series
 Reimann Zeta function
 Number theory
Gamma and Beta function

3. Beta Function
 The first Eulerian function is known as Beta function
 Beta Function is denoted by β(m,n) and defined by

4. Evaluation of Beta function
 By definition
 Integrating by parts, considering as first function, we
get

5.  Integrating again by parts, we get
 Continuing the process of Integrating by parts,
We obtain

6. Properties of beta function
 Symmetry property
 Transformation of beta function

7. Gamma function
 The Second Eulerian function is known as Gamma
function
 it is denoted by Г(n) and defined by

8. Evaluation of Gamma function
 By definition
 Integratin g by parts, considering as first
function, we get

9.  Integrating again by by parts
 If n is the postive integer,then proceeding as
above properly, We get

10. Properties of Gamma Function

11. Relation between Beta and Gamma
Function

12. References
 Arfken G. and Weber H.J., Mathematical
Methods for Physicists, Academic Press
(2005).
 Rajput B. S., Mathematical Physics, Pragati
Prakashan (2002).
 Boas M.L. Mathematical Methods in the
Physical Sciences, John Wiley & Sons, New
York (1983).
 Harper C. Analytical Mathematics in Physics,
Prentice Hall (1999).

13. Thank You