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
- <math>\Gamma(x)=\int_0^\infty t^{x-1} e^{-t}\,dt</math>
as the only function f on the interval x > 0 that simultaneously has the three properties
- <math>f(1)=1\mbox{,} \,</math> and
- <math>f(x+1)=xf(x)\ \mbox{for}\ x>0, \,</math> and
- <math>\log f \,</math> is a convex function. (That is <math>f \,</math> 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.
References
- Eric W. Weisstein, Bohr-Mollerup Theorem at 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.
