PlanetPhysics/Gamma Function

The gamma function is

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

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

The Gamma function satisfies

$\Gamma (x+1)=x\Gamma (x)$

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

$\Gamma (n)=(n-1)!$

Some values of the gamma function for small arguments are:

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

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

$\Gamma (n+f)$

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

${\begin{matrix}\Gamma (n+f)&=&(n+f-1)\Gamma (n+f-1)\\&=&(n+f-1)(n+f-2)\Gamma (n+f-2)\\&\vdots &\\&=&(n+f-1)(n+f-2)\cdots (f)\Gamma (f)\end{matrix}}$

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

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

where $\gamma$ is Euler's constant, and the functional equation

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

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