Bohr–Mollerup theorem
Theorem in complex analysis / From Wikipedia, the free encyclopedia
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In mathematical analysis, the Bohr–Mollerup theorem[1][2] is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup.[3] The theorem characterizes the gamma function, defined for x > 0 by
as the only positive function f , with domain on the interval x > 0, that simultaneously has the following three properties:
- f (1) = 1, and
- f (x + 1) = x f (x) for x > 0 and
- f is logarithmically convex.
A treatment of this theorem is in Artin's book The Gamma Function,[4] which has been reprinted by the AMS in a collection of Artin's writings.[5]
The theorem was first published in a textbook on complex analysis, as Bohr and Mollerup thought it had already been proved.[3]
The theorem admits a far-reaching generalization to a wide variety of functions (that have convexity or concavity properties of any order).[6]