Bohr–Mollerup theorem Information & Bohr–Mollerup theorem Links at HealthHaven.com

In mathematical analysis, the Bohr–Mollerup theorem is named after the Danish mathematicians Harald Bohr and Johannes Mollerup, who proved it. The theorem characterizes the gamma function, defined for x > 0 by

$\Gamma(x)=\int_0^\infty t^{x-1} e^{-t}\,dt$

as the only function f on the interval x > 0 that simultaneously has the three properties

$f(1)=1\mbox{,} \,$ and
$f(x+1)=xf(x)\ \mbox{for}\ x>0, \,$ and
$\log f \,$ is a convex function. (That is $f \,$ is logarithmically convex.)

That log f is convex is often expressed by saying that f is log-convex, i.e., a log-convex function is one whose logarithm is convex.

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.

Interestingly, the theorem was first published in a textbook on complex analysis, as Bohr and Mollerup had thought it proven already.

[edit] References

Weisstein, Eric W., "Bohr-Mollerup Theorem" from MathWorld.
Proof of Bohr-Mollerup theorem on PlanetMath
Proof of Bohr-Mollerup theorem on PlanetMath
Artin, Emil (1964). The Gamma Function. Holt, Rinehart, Winston.
Rosen, Michael (2006). Exposition by Emil Artin: A Selection. American Mathematical Society.
Mollerup, J., Bohr, H. (1922). Lærebog i Kompleks Analyse vol. III, Copenhagen. (Textbook in Complex Analysis)

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