26
Dec 21

## Gamma function

### Gamma function

The gamma function and gamma distribution are two different things. This post is about the former and is a preparatory step to study the latter.

Definition. The gamma function is defined by

$\Gamma \left( t\right) =\int_{0}^{\infty }x^{t-1}e^{-x}dx,\ t> 0.$

The integrand $f(t)=x^{t-1}e^{-x}$ is smooth on $\left( 0,\infty \right) ,$ so its integrability is determined by its behavior at $\infty$ and $0$. Because of the exponent, it is integrable in the neighborhood of $\infty .$ The singularity at $0$ is integrable if $t>0.$ In all calculations involving the gamma function one should remember that its argument should be positive.

## Properties

1) Factorial-like property. Integration by parts shows that

$\Gamma \left( t\right) =-\int_{0}^{\infty }x^{t-1}\left( e^{-x}\right) ^{\prime }dx=-x^{t-1}e^{-x}|_{0}^{\infty }+\left( t-1\right) \int_{0}^{\infty }x^{t-2}e^{-x}dx$

$=\left( t-1\right) \Gamma \left( t-1\right)$ if $t>1.$

2) $\Gamma \left( 1\right) =1$ because $\int_{0}^{\infty }e^{-x}dx=1.$

3) Combining the first two properties we see that for a natural $n$

$\Gamma \left( n+1\right) =n\Gamma ( n) =...=n\times \left( n-1\right) ...\times 1\times \Gamma \left( 1\right) =n!$

Thus the gamma function extends the factorial to non-integer $t>0.$

4) $\Gamma \left( 1/2\right) =\sqrt{\pi }.$

Indeed, using the density $f_{z}$ of the standard normal $z$ we see that

$\Gamma \left( 1/2\right) =\int_{0}^{\infty }x^{-1/2}e^{-x}dx=$

(replacing $x^{1/2}=u$)

$=\int_{0}^{\infty }\frac{1}{u}e^{-u^{2}}2udu=2\int_{0}^{\infty }e^{-u^{2}}du=\int_{-\infty }^{\infty }e^{-u^{2}}du=$

(replacing $u=z/\sqrt{2}$)

$=\frac{\sqrt{\pi }}{\sqrt{2\pi }}\int_{-\infty }^{\infty }e^{-z^{2}/2}dz= \sqrt{\pi }\int_{R}f_{z}\left( t\right) dt=\sqrt{\pi }.$

Many other properties are not required in this course.