# gamma function

generated by

## Definitions

 $\Gamma(z)=\lim_{n\to\infty}\frac{n^{z}n!}{\prod_{k=0}^{n}(z+k)}.$

However, it is now more commonly defined by

 $\Gamma(z)=\int_{0}^{\infty}\!e^{-t}t^{z-1}\,dt$

for $z\in\mathbb{C}$ with $\operatorname{Re}(z)>0$, and by analytic continuation for the rest of the complex plane, except for the non-positive integers (where it has simple poles   ).

 $\Gamma(z)=\frac{e^{-\gamma z}}{z}\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right% )^{-1}e^{z/n},$

where $\gamma$ is Euler’s constant.

## Functional equations

The gamma function satisfies the functional equation

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

except when $z$ is a non-positive integer. As $\Gamma(1)=1$, it follows by induction  that

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

for positive integer values of $n$.

Another functional equation satisfied by the gamma function is

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

for non-integer values of $z$.

## Approximate values

The gamma function for real $z$ looks like this: (generated by GNU Octave and gnuplot)

It can be shown that $\Gamma(1/2)=\sqrt{\pi}$. Approximate values of $\Gamma(x)$ for some other $x\in(0,1)$ are:

 $\begin{array}[]{cc}\Gamma(1/5)\approx 4.5908&\Gamma(1/4)\approx 3.6256\\ \Gamma(1/3)\approx 2.6789&\Gamma(2/5)\approx 2.2182\\ \Gamma(3/5)\approx 1.4892&\Gamma(2/3)\approx 1.3541\\ \Gamma(3/4)\approx 1.2254&\Gamma(4/5)\approx 1.1642\end{array}$

If the value of $\Gamma(x)$ is known for some $x\in(0,1)$, then one may calculate the value of $\Gamma(n+x)$ for any integer $n$ by making use of the formula $\Gamma(z+1)=z\Gamma(z)$. We have

 $\displaystyle\Gamma(n+x)$ $\displaystyle=$ $\displaystyle(n+x-1)\Gamma(n+x-1)$ $\displaystyle=$ $\displaystyle(n+x-1)(n+x-2)\Gamma(n+x-2)$ $\displaystyle\vdots$ $\displaystyle=$ $\displaystyle(n+x-1)(n+x-2)\cdots(x)\Gamma(x)$

which is easy to calculate if we know $\Gamma(x)$.

## References

• 1 Julian Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003. (Chapter 6 is about the gamma function.)
Title gamma function GammaFunction 2013-03-22 12:00:39 2013-03-22 12:00:39 yark (2760) yark (2760) 44 yark (2760) Definition msc 30D30 msc 33B15 gamma-function $\Gamma$-function Euler’s gamma function Euler’s gamma-function Euler’s $\Gamma$-function BohrMollerupTheorem TableOfLaplaceTransforms