proof of Bohr-Mollerup theorem
The inequality follows from Hölder’s inequality, where and .
Now we show that the 3 conditions uniquely determine a function. By condition 2, it suffices to show that the conditions uniquely determine a function on .
Let be a function satisfying the 3 conditions, and .
and by log-convexity of , .
Similarly gives .
Combining these two we get
and by using condition 2 to express in terms of we find
Now these inequalities hold for every positive integer and the terms on the left and right side have a common limit () so we find this determines .
As a corollary we find another expression for .
In fact, this equation, called Gauß’s product, goes for the whole complex plane minus the negative integers.
|Title||proof of Bohr-Mollerup theorem|
|Date of creation||2013-03-22 13:18:14|
|Last modified on||2013-03-22 13:18:14|
|Last modified by||lieven (1075)|