MY MATEX NOTES

In this post we prove the following infinite sum Click here for the proof. We relied in a few previous established results: Proved here Both proved here . And Proved here .

This is a short article to prove the following result To this end we will rely on some previous established results, namely: proved here, and proved here . As a corollary of our goal series we also obtain this nice series Click here  for the proof of our main series.

In this article we want to prove the following three infinte series: click here for the proof.

In this entry we will evaluate symmetric infinite sums of the type To this end we will rely on the previous established result ( proved here ) Click here for the proof.

In this blog entry we will prove the following three beautifull infinte series Click here for the proof.

In this entry we present a proof via contour integration for the following alternate infinite series Click here for the proof.

In this entry we present a second proof via contour integration for the series (proved previously here ) Click here for the proof.

In this article we prove two remarkable results due to Ramanujan, namely and To this end we rely on the functional equation that the Dedekind´s eta function obeys, previously proved ( here ): Click here for the proof.

In this article we prove the following Ramanujan series To this end we will rely on the residues theorem.  Click here  for the proof of the Ramanujan´s series

We want to prove the following transformation formula that appears in this twitter post Click here to see the proof . We used the result proved here .

          Today we will evaluate the following infinite sum which corresponds to the derivative of Dirichlet eta function @ 1 As a bonus we will compute the following integral Lets first introduce a lemma: Lemma 1: (1) Proof: Claim: (2) If we let     in (1) we get (3) Lets now recall the Euler Maclaurin Formula (proved here ) to estimate the last two sums above (4) Choosing    and         in (4), we get for the first sum : (5) And for the second sum (6) Recall also the integral representation of the Stiltjies constant (shown here ): (7) letting in (7) we obtain (8) Now, plugging (5) and (6) back in (3) and letting we obtain We can now use (2) to calculate the following integral Proof:

          In today´s post we will derive a functional equation for the Hurwitz Zeta Function, namely: To this end we introduce the periodic zeta function defined by the following expression: (1) Theorem 1: (2) for and Proof: First recall the previously proved fourier expansion (see here ) (3) for and Now lets expand (2) by means of Euler´s formula Lemmma 1: Let h and k be two integers, , then for (4) Proof: To prove this equality used in the proof above Lets expand first it´s L.H.S. We used the fact that for m an integer Now, lets expand it´s R.H.S. and show that they are equal to each other Theorem 2: If k and h are integers with , then for all s we have Proof: Plugging (4) in (2) we obtain: