III.31 The Gamma Function, Ben Green: Page 214.

Extension of the integer factorial definition to the real and complex numbers.

There are multiple possible extensions.

The one chosen is such that f(x + 1) = x f(x), f(1) = 1 and log(f) is convex.  This yields the gamma function as the definition of the extension, where,

G(s) = ò(0,¥) xs-1e-xdxThere are interesting formulas involving the gamma function including Sterling’s formula. 

Leave a Reply

You must be logged in to post a comment.