伽马函数的另一种表征

The gamma function Γ(z) extends the factorial function from integers to complex numbers. (Technically, Γ(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) = Γ(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 if f 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) = Γ(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 Γ(z) that

|Γ(z)| ≤ |Γ(Re z)|

for z 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.