Gamma functions
Gamma function, generalization of the factorial function to nonintegral values, introduced by the Swiss mathematician Leonhard Euler in the 18th century.
The gamma function is defined as
$$ \Gamma(x) = \int\limits_0^{\infty} t^{x-1}e^{-1}dt $$
Properties of Gama function:
$$ 1. \ \Gamma(\alpha+1)= \alpha\Gamma(\alpha) $$
$$ 2. \ \Gamma(n)=(n-1)!, \ \text{for} \ n=1,2,3, \cdots $$
$$ 3. \ \Gamma\left(\frac 12\right)= \sqrt {\pi} $$
$$ 4. \ \Gamma(z)\Gamma(1-z)=\frac {\pi}{\sin z\pi} $$
$$ 5. \ \Gamma(x)\Gamma\left(x+\frac 12\right)= \frac {\sqrt {\pi}}{2^{2x-1}}\Gamma(2x) $$
Example:
$$ \text{Use the properties of Γ to show that} \ \Gamma\left(\frac 32\right)= \frac {\sqrt {\pi}}{2} $$
Solution:
$$ \Gamma\left(\frac 32\right)= \Gamma\left(\frac 12 + 1\right) $$
$$ \Gamma\left(\frac 32\right)=\frac 12\Gamma\left(\frac 12\right) $$
$$ \Gamma\left(\frac 32\right)=\frac {\sqrt {\pi}}{2} $$