Lemma 1. Let be a function with continuous derivative in its domain.
By the Fundamental Theorem of Calculus:
Now integrating by parts: and Lemma 1 follows.
Lemma 2. We have:
in Lemma 1:
Now note that the integral in the RHS converges as
- this in fact shows the existance of the gamma constant- then by definition of the gamma constant:
in Lemma 1
Therefore: Lemma 3.
we are done Proof :
be the integral then letting
Now note that:
Applying Lemma 2