nLab
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 $x\ne -1,-2,\dots$, we define $\Gamma (x)$ by the formula

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

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

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

is the shifted factorial. It easily follows that $\Gamma (n+1)=n!\mathrm{for}\mathrm{natural}\mathrm{numbers}$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 $x=0,-1,-2,\dots$.

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

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

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

It also satisfies a reflection formula, due to Euler:

$$\Gamma (x)\Gamma (1-x)=\frac{\pi }{\sin (\pi x)}.$$\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 $(0,\mathrm{\infty })$ is the unique function $f$ such that $f(x+1)=xf(x)$, $f(1)=1$, and $\log f$ is convex.

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

• Product representation:

  $$\frac{1}{\Gamma (x)}=x{e}^{\gamma x}\prod _{n=1}^{\mathrm{\infty }}(1+\frac{x}{n})e^{-x/n}$$\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:

  $$\Gamma (x)={\int }_0^{\mathrm{\infty }}{t}^x{e}^{-t}\frac{dt}{t}.$$\Gamma(x) = \int_{0}^{\infty} t^x e^{-t} \frac{d t}{t}.

References

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