If we consider the integral , it is once seen to be an infinite and improper integral. This integral is infinite because the upper limit of integration is infinite and it is improper because is a point of infinite discontinuity of the integrand, if , where is either real number or real part of a complex number. This integral is known as Euler’s Integral. This is of a great importance in mathematical analysis and calculus. The result, i.e., integral, is defined as a new function of real number , as .
Definitions for Gamma Function
Let be any positive real number, then we can integrate the Eulerian Integral by assuming as first function and as second function, integrating it by parts.
After a little work, one might get
This is defined as gamma function of (i.e., ) and .
If is a positive integer, then we can write .
This definition is not defined for Gamma Function for negative numbers and zero. The second definition of Gamma Function is given terms of Euler’s infinite limit
, where be either real or complex number.
Third definition of gamma function is given in terms of Weierstrass’s infinite product, as for any number is,
; where is a constant, called Euler-Mascheroni Constant and its value is approximately .
All the three definitions defined above are equivalent to each other.
Important Properties of Gamma Functions
1. If is a positive integer:
2. If is a negative integer or zero:
3. If is a non-integer number, then exists and has a finite value.