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Gamma function

The function

The Γ\Gamma function

Idea and Definition

Leonhard Euler solved the problem of finding a function of a continuous variable xx which for integer values of x=nx=n agrees with the factorial function nn!n\mapsto n!. The gamma function is a shift by one of the solution to this problem.

For a complex variable x1,2,x\neq -1,-2,\ldots, we define Γ(x)\Gamma(x) by the formula

Γ(x)=lim kk!k x1(x) k \Gamma(x) = \lim_{k\to \infty} \frac{k! \cdot k^{x-1}}{(x)_k}

where (x) 0=1(x)_0 =1 and for positive integer k=1,2,k = 1,2,\ldots,

(x) k=x(x+1)(x+2)(x+k1) (x)_k = x (x+1) (x+2)\cdots (x+ k-1)

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

Properties

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

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

(1)Γ(x+1)=xΓ(x) \Gamma(x+1) \;=\; x\,\Gamma(x)

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

It also satisfies a reflection formula, due to Euler:

Γ(x)Γ(1x)=πsin(πx).\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}.

Proposition

(Gauss multiplication formula)

For

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

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

the Gamma function Γ()\Gamma(-) satisfies

j=0N1Γ(z+jN)=(2π) 12(N1)N 12NzΓ(Nz). \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 interval (0,)(0, \infty) is the unique function ff such that f(x+1)=xf(x)f(x+1) = x f(x), f(1)=1f(1) = 1, and logf\log f is convex.

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

  • Product representation:

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

    where γ\gamma is Euler's constant.

  • Integral representation:

    Γ(x)= 0 t xe tdtt.\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.

See also:

Last revised on June 1, 2021 at 08:44:42. See the history of this page for a list of all contributions to it.