There are lots (an infinitude) of smooth functions that coincide with f(n)=n! on the integers. Is there a simple reason why Euler’s Gamma function Γ(z)=∫∞0tz−1e−tdt is the “best”? In particular, I’m looking for reasons that I can explain to first-year calculus students.
The Bohr–Mollerup theorem shows that the gamma function is the only function that satisfies the properties
- f(x+1)=xf(x) for every x≥0;
- logf is a convex function.
The condition of log-convexity is particularly important when one wants to prove various inequalities for the gamma function.
By the way, the gamma function is not the only meromorphic function satisfying
with no zeroes and no poles other than the points z=−n, n=0,1,2…. There is a whole family of such functions, which, in general, have the form
where g(z) is an entire function such that
(γ is Euler’s constant). The gamma function corresponds to the simplest choice
Edit: corrected index in the product.