## Gamma functions- beta gamma Formulas

### 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:

$$\Gamma\left(\frac 32\right)= \Gamma\left(\frac 12 + 1\right)$$
$$\Gamma\left(\frac 32\right)=\frac 12\Gamma\left(\frac 12\right)$$
$$\Gamma\left(\frac 32\right)=\frac {\sqrt {\pi}}{2}$$