proof of multiplication formula for gamma function

Define the functionMathworldPlanetmath f as


By the functional equation of the gamma functionDlmfDlmfMathworldPlanetmath,


Hence f is a periodic functionMathworldPlanetmath of z. However, for large values of z, we can apply the Stirling approximation formula to conclude


Note that




Hence, f(z)=(2π)(n-1)/2n1/2+O(z-1). Now, the only way for a function to be periodic and have a definite limit is for that function to be constant. Therefore, f(z)=(2π)(n-1)/2n1/2. Writing out the definition of f and rearranging gives the multiplication formula.

Title proof of multiplication formula for gamma function
Canonical name ProofOfMultiplicationFormulaForGammaFunction
Date of creation 2013-03-22 14:44:10
Last modified on 2013-03-22 14:44:10
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Proof
Classification msc 33B15
Classification msc 30D30