Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, defined for x > 0 by
as the only function f on the interval x > 0 that simultaneously has the three properties
An elegant treatment of this theorem is in Artin's book The Gamma Function, which has been reprinted by the AMS in a collection of Artin's writings.
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.
Let Γ(x) be a function with the assumed properties established above: Γ(x + 1) = xΓ(x) and log(Γ(x)) is convex, and Γ(1) = 1. From Γ(x + 1) = xΓ(x) we can establish
The purpose of the stipulation that Γ(1) = 1 forces the Γ(x + 1) = xΓ(x) property to duplicate the factorials of the integers so we can conclude now that Γ(n) = (n − 1)! if n ∈ N and if Γ(x) exists at all. Because of our relation for Γ(x + n), if we can fully understand Γ(x) for 0 < x ≤ 1 then we understand Γ(x) for all values of x.
The slope of a line connecting two points (x1, log(Γ (x1))) and (x2, log(Γ (x2))), call it S(x1, x2), is monotonically increasing in each argument with x1 < x2 since we have stipulated log(Γ(x)) is convex. Thus, we know that
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