Raising the bar

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

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

The integrand $latex f(t)=x^{t-1}e^{-x}$ is smooth on $latex \left( 0,\infty \right) ,$ so its integrability is determined by its behavior at $latex \infty $ and $latex 0$. Because of the exponent, it is integrable in the neighborhood of $latex \infty .$ The singularity at $latex 0$ is integrable if $latex t>0.$ 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

$latex \Gamma( t) =-\int_0^\infty x^{t-1}\left( e^{-x}\right)^\prime dx=-x^{t-1}e^{-x}|_0^\infty+(t-1)\int_0^{\infty }x^{t-2}e^{-x}dx$

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

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

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

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

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

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

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

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

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

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

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

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

Many other properties are not required in this course.