Bohr–Mollerup theorem

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

* $f(1)=1mbox\{,\}\; ,$ 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 beenreprinted by the AMS in a collection of Artin's writings.

References

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* cite book |last= Artin |first= Emil |title= The Gamma Function
year= 1964 |publisher= Holt, Rinehart, Winston
* cite book |last= Rosen |first= Michael |title= Exposition by Emil Artin: A Selection
year= 2006 |publisher= American Mathematical Society