# The $\Gamma$ function

## Idea and Definition

Leonhard Euler solved the problem of finding a function of a continuous variable $x$ which for integer values of $x=n$ agrees with the factoriel function $n\mapsto n!$. In fact, gamma function is a shift by one of the solution of this problem.

For a complex variable $x\neq -1,-2,\ldots$, we define $\Gamma(x)$ by the formula

$\Gamma(x) = \lim_{k\to \infty} \frac{k! k^{x-1}}{(x)_k}$

where $(x)_0 =1$ and for positive integer $k = 1,2,\ldots$,

$(x)_k = x (x+1) (x+2)\cdots (x+ k-1)$

is the “Pochhammer symbol” (or rising factorial). It easily follows that $\Gamma(n+1) = n!$ for natural numbers $n = 0, 1, 2, \ldots$.

## Properties

As a function of a complex variable, the Gamma function $\Gamma(x)$ is a meromorphic function with simple poles at $x = 0, -1, -2, \ldots$.

Extending the recursive definition of the ordinary factorial function, the Gamma function satisfies the following translation formula:

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

away from $x = 0, -1, -2, \ldots$.

It also satisfies a reflection formula, due to Euler:

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

Quite remarkably, the Gamma function (this time as a function of a real variable) is uniquely characterized in the following theorem:

###### Theorem (Bohr-Mollerup)

The restriction of the Gamma function to the interval $(0, \infty)$ is the unique function $f$ such that $f(x+1) = x f(x)$, $f(1) = 1$, and $\log f$ is convex.

A number of other representations of the Gamma function are known and frequently utilized, e.g.,

• Product representation:

$\frac1{\Gamma(x)} = x e^{\gamma x} \prod_{n=1}^{\infty} (1 + \frac{x}{n})e^{-x/n}$

where $\gamma$ is Euler's constant?.

• Integral representation:

$\Gamma(x) = \int_{0}^{\infty} t^x e^{-t} \frac{d t}{t}.$
• completed Riemann zeta function?

## References

• George Andrews, Richard Askey, Ranjan Roy, Special Functions. Encyclopedia of Mathematics and Its Applications 71, Cambridge University Press, 1999.

Revised on June 26, 2015 13:29:03 by Noam Zeilberger (80.215.169.3)