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gamma function

(Definition)

Introduction

The gamma function can be thought of as the natural way to generalize the concept of the factorial to non-integer arguments.

Leonhard Euler came up with a formula for such a generalization in 1729. At around the same time, James Stirling independently arrived at a different formula, but was unable to show that it always converged. In 1900, Charles Hermite showed that the formula given by Stirling does work, and that it defines the same function as Euler's.

Definitions

Euler's original formula for the gamma function was

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

However, it is now more commonly defined by

$\displaystyle \Gamma(z) = \int_0^\infty \! e^{-t} t^{z-1} dt$

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).

Another equivalent definition is

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

where $\gamma$ is Euler's constant.

Functional equations

The gamma function satisfies the functional equation

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

except when

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

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

for positive integer values of

Another functional equation satisfied by the gamma function is

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

for non-integer values of

Approximate values

The gamma function for real looks like this:

$\includegraphics[scale=1]{gammafunc.eps}$

(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:

$\begin{displaymath} \begin{array}{cc} \Gamma(1/5) \approx 4.5908 & \Gamma(1/4) \... ...ma(3/4) \approx 1.2254 & \Gamma(4/5) \approx 1.1642 \end{array}\end{displaymath}$

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 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 $\Gamma(x)$ .

Bibliography

1: Julian Havil, Gamma: Exploring Euler's Constant, Princeton University Press, 2003. (Chapter 6 is about the gamma function.)

"gamma function" is owned by yark. [ full author list (2) | owner history (1) ]

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See Also: Bohr-Mollerup theorem

Other names:

gamma-function, $\Gamma$ -function, Euler's gamma function, Euler's gamma-function, Euler's $\Gamma$ -function

Attachments:: multiplication formula for gamma function (Theorem) by rspuzio
incomplete gamma function (Definition) by rspuzio
digamma and polygamma function (Definition) by rspuzio
Euler reflection formula (Theorem) by rm50
evaluating the gamma function at 1/2 (Derivation) by CWoo
analytic continuation of gamma function (Derivation) by pahio
Hankel contour integral (Result) by perucho
values of gamma function for small positive real values (Data Structure) by PrimeFan

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Cross-references: real, positive, induction, functional equation, Euler's constant, simple poles, integers, complex plane, analytic continuation, function, factorial
There are 44 references to this entry.

This is version 40 of gamma function, born on 2001-11-17, modified 2007-10-22.
Object id is 955, canonical name is GammaFunction.
Accessed 24889 times total.

Classification:

AMS MSC:	33B15 (Special functions :: Elementary classical functions :: Gamma, beta and polygamma functions)
	30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)

Pending Errata and Addenda

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