A lesser-known characterization of the gamma function

The gamma function (z) extends the factorial function from integers to complex numbers. (Technically, (z + 1) extends factorial.) There are other ways to extend the factorial function, so what makes the gamma function the right choice?

The most common answer is the Bohr-Mollerup theorem. This theorem says that if f: (0, ) (0, ) satisfies

1. f(x + 1) = x f(x)
2. f(1) = 1
3. log f is convex

then f(x) = (x). The theorem applies on the positive real axis, and there is a unique holomorphic continuation of this function to the complex plane.

But the Bohr-Mollerup theorem is not the only theorem characterizing the gamma function. Another theorem was by Helmut Wielandt. His theorem says that iff is holomorphic in the right half-plane and

1. f(z + 1) = z f(z)
2. f(1) = 1
3. f(z) is bounded for {z: 1 Re z 2}

then f(x) = (x). In short, Weilandt replaces the log-convexity for positive reals with the requirement that f is bounded in a strip of the complex plane.

You might wonder what is the bound alluded to in Wielandt's theorem. You can show from the integral definition of (z) that

|(z)| |(Re z)|

forz in the right half-plane. So the bound on the complex strip {z: 1 Re z 2} equals the bound on the real interval [1, 2], which is 1.

The post A lesser-known characterization of the gamma function first appeared on John D. Cook.