The function

# 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 factorial function $n\mapsto n!$. The gamma function is a shift by one of the solution to this problem.

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

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

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:

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

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

It also satisfies a reflection formula, due to Euler:

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

###### Proposition

(Gauss multiplication formula)

For

• any positive integer $N \in \mathbb{N}_+$,

• any $z \in \mathbb{R} \setminus \{0, -1/N, -2/N, \cdots\}$,

the Gamma function $\Gamma(-)$ satisfies

$\underoverset {j = 0} {N-1} {\prod} \Gamma \left( z + \tfrac{j}{N} \right) \;=\; (2 \pi)^{ \tfrac{1}{2}(N-1) } \cdot N^{ \tfrac{1}{2} - N z } \cdot \Gamma( N z ) \,.$

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 open interval $(0, \infty)$ is the unique function $f$ such that

1. $f(x+1) = x f(x)$,

2. $f(1) = 1$,

3. $\log f$ is convex.

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

• Product representation:

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

where $\gamma$ is Euler's constant.

• Integral representation:

$\Gamma(x) \;=\; \int_{0}^{\infty} t^x e^{-t} \frac{d t}{t} \,.$

## References

• Emil Artin, Einführung in die Theorie der Gammafunktion, Hamburger Mathematische Einzelschriften

l. Heft, Verlag B. G. Teubner, Leipzig (1931)

English translation by Michael Butler: The Gamma Function, Holt, Rinehart and Winston (1931) [pdf]

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