Perhaps a number of you are familiar with the solution to this integral. if not, see if you can work it out.
If you do them individually, you just get
But that is certainly not the solution.
It has the same solution as:
I have worked on this integral for about two weeks and have finally come up with a solution
First split it into
Then make the sub on the second integral turning it into
Integration by parts on both gives
Now consider the gamma function
Well since the Gamma function is uniformly convergent for all values of x that make it converge we have that
The value of Gamma' evaluated at one...I cannot prove..haha
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:
Lemma.
Proof
By definition:
Note that: thus:
We get: Consider that: then
Exchange the summation order ( this can be justified working with the remainder, see here):
Proof.
Letting we have:
Now use the Power series expansion of :
So by Lemma 3of the previous post: And finally, by the Lemma we've just seen: