**Lemma 1. **Let be a function with continuous derivative in its domain.

Then where

__Proof__

By the Fundamental Theorem of Calculus:

So:

Now integrating by parts: and *Lemma 1* follows.

**Lemma 2. **We have:

__Proof __

First take

in

*Lemma 1: *We get

Thus:

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:

Now take

with

in

*Lemma 1 *:

Thus:

and then

Therefore:

**Lemma 3. ** :

*Proof: * now since

we are done

**Proof : **
Let

be the integral then letting

we get

Now note that:

(

)

Thus:

Then:

Applying

*Lemma 2*: