gamma function


generated by


The gamma functionDlmfDlmfMathworldPlanetmath can be thought of as the natural way to generalize the conceptMathworldPlanetmath of the factorial to non-integer argumentsPlanetmathPlanetmath.

Leonhard Euler ( came up with a formulaMathworldPlanetmathPlanetmath for such a generalizationPlanetmathPlanetmath 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 functionMathworldPlanetmath as .



However, it is now more commonly defined by


for z with Re(z)>0, and by analytic continuation for the rest of the complex plane, except for the non-positive integers (where it has simple polesMathworldPlanetmathPlanetmath).

Another equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath definition is


where γ is Euler’s constant.

Functional equations

The gamma function satisfies the functional equation


except when z is a non-positive integer. As Γ(1)=1, it follows by inductionMathworldPlanetmath that


for positive integer values of n.

Another functional equation satisfied by the gamma function is


for non-integer values of z.

Approximate values

The gamma function for real z looks like this:

(generated by GNU Octave and gnuplot)

It can be shown that Γ(1/2)=π. Approximate values of Γ(x) for some other x(0,1) are:


If the value of Γ(x) is known for some x(0,1), then one may calculate the value of Γ(n+x) for any integer n by making use of the formula Γ(z+1)=zΓ(z). We have

Γ(n+x) = (n+x-1)Γ(n+x-1)
= (n+x-1)(n+x-2)Γ(n+x-2)
= (n+x-1)(n+x-2)(x)Γ(x)

which is easy to calculate if we know Γ(x).


  • 1 Julian Havil, Gamma: Exploring Euler’s Constant, Princeton University Press, 2003. (Chapter 6 is about the gamma function.)
Title gamma function
Canonical name GammaFunction
Date of creation 2013-03-22 12:00:39
Last modified on 2013-03-22 12:00:39
Owner yark (2760)
Last modified by yark (2760)
Numerical id 44
Author yark (2760)
Entry type Definition
Classification msc 30D30
Classification msc 33B15
Synonym gamma-function
Synonym Γ-function
Synonym Euler’s gamma function
Synonym Euler’s gamma-function
Synonym Euler’s Γ-function
Related topic BohrMollerupTheorem
Related topic TableOfLaplaceTransforms