The Γ-function and Mellin’s Theorem

Abstract

The factorial function

$$m! = \,1 \cdot 2...m$$

(18.1)

is only defined for natural numbers m. It fits into most arguments to put also 0! = 1, as we have done throughout the previous chapters. The problem to extend the factorial function to other values has of course no unique answer, for if we would have found any definition for x! for continuously varying x any other definition differing from it by a function vanishing at all non-negative integers (e.g. sin πx) will solve the interpolation problem just as well. One has therefore to look for some plausible reason to choose a more or less natural extension of the function m!