Gamma Function

If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , 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 $t=0$ is a point of infinite discontinuity of the integrand, if $a<1$ , where $a$ 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 $a$ , as $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

Definitions for Gamma Function
Let $a$ be any positive real number, then we can integrate the Eulerian Integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ by assuming $t^{a-1}$ as first function and $e^{-t}$ as second function, integrating it by parts.
After a little work, one might get
$I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt=(a-1)(a-2) \ldots 2\cdot 1$ .
This is defined as gamma function of $a$ (i.e., $\Gamma (a)$ ) and $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt =(a-1)(a-2) \ldots 2 \cdot 1$ .
If $a$ is a positive integer, then we can write $\Gamma a=(a-1)!$ .
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
$\Gamma (a)=\displaystyle{\lim_{m \to \infty}} \dfrac{1\cdot 2 \cdot 3 \cdots m}{a(a+1)(a+2) \ldots (a+m)} m^a$ , where $a$ be either real or complex number.

Third definition of gamma function is given in terms of Weierstrass’s infinite product, as $\Gamma (a)$ for any number $a$ is,
$\dfrac{1}{\Gamma a} =a e^{a \gamma} \displaystyle{\prod_{m=1}^{\infty}} \left({1+\frac{m}{a}}\right) e^{-a/m}$ ; where $\gamma =\lim_{n \to {\infty}} \left({1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}-\log n}\right)$ is a constant, called Euler-Mascheroni Constant and its value is approximately $0.5772157$ .
All the three definitions defined above are equivalent to each other.

Important Properties of Gamma Functions
1. If $n$ is a positive integer: $\Gamma (n)=(n-1)!=(n-1)(n-2)(n-3)\ldots 2\cdot 1$
2. If $n$ is a negative integer or zero: $\Gamma (n)=\infty$
3. If $n$ is a non-integer number, then $\Gamma (n)$ exists and has a finite value.