PlanetPhysics/Gamma Function

The gamma function is

$\mathrm{\Gamma }(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-t}{t}^{x-1}dt$

where $x\in ℂ\setminus \{0,-1,-2,\dots \}$.

The Gamma function satisfies

$\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$

Therefore, for integer values of $x=n$,

$\mathrm{\Gamma }(n)=(n-1)!$

Some values of the gamma function for small arguments are:

$\begin{array}{cc}\mathrm{\Gamma }(1/5)=4.5909& \mathrm{\Gamma }(1/4)=3.6256\\ \mathrm{\Gamma }(1/3)=2.6789& \mathrm{\Gamma }(2/5)=2.2182\\ \mathrm{\Gamma }(3/5)=1.4892& \mathrm{\Gamma }(2/3)=1.3541\\ \mathrm{\Gamma }(3/4)=1.2254& \mathrm{\Gamma }(4/5)=1.1642\end{array}$

and the ever-useful $\mathrm{\Gamma }(1/2)=\sqrt{\pi }$. These values allow a quick calculation of

$\mathrm{\Gamma }(n+f)$

Where $n$ is a natural number and $f$ is any fractional value for which the Gamma [[../Bijective/|function's]] value is known. Since $\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$, we have

$\begin{array}{ccc}\mathrm{\Gamma }(n+f)& =& (n+f-1)\mathrm{\Gamma }(n+f-1)\\ & =& (n+f-1)(n+f-2)\mathrm{\Gamma }(n+f-2)\\ & ⋮& \\ & =& (n+f-1)(n+f-2)\cdots (f)\mathrm{\Gamma }(f)\end{array}$

Which is easy to calculate if we know $\mathrm{\Gamma }(f)$.

The gamma function has a meromorphic continuation to the entire complex plane with poles at the non-positive integers. It satisfies the product [[../Formula/|formula]] $\mathrm{\Gamma }(z)=\frac{e^{-\gamma z}}{z}\prod _{n=1}^{\mathrm{\infty }}{(1+\frac{z}{n})}^{-1}{e}^{z/n}$

where $\gamma$ is [[../EulerConstant/|Euler's constant]], and the functional equation

$\mathrm{\Gamma }(z)\mathrm{\Gamma }(1-z)=\frac{\pi }{\sin \pi z}.$

This entry is a derivative of the gamma function article from PlanetMath. Author of the orginial article: akrowne. History page of the original is here