evaluating the gamma function at 1/2

In the entry on the gamma functionDlmfDlmfMathworldPlanetmath it is mentioned that Γ(1/2)=π. In this entry we reduce the proof of this claim to the problem of computing the area under the bell curve. First note that by definition of the gamma function,

Γ(1/2) =0e-xx-1/2𝑑x

Performing the substitution u=x, we find that du=12xdx, so


where the last equality holds because e-u2 is an even functionMathworldPlanetmath. Since the area under the bell curve is π, it follows that Γ(1/2)=π.

Title evaluating the gamma function at 1/2
Canonical name EvaluatingTheGammaFunctionAt12
Date of creation 2013-03-22 16:57:13
Last modified on 2013-03-22 16:57:13
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 4
Author CWoo (3771)
Entry type Derivation
Classification msc 30D30
Classification msc 33B15
Related topic AreaUnderGaussianCurve
Related topic LaplaceTransformOfPowerFunction