MY MATEX NOTES

            In this entry I´ll share the draft of some chapters of my ongoing personal project of writing a book. The chapters here serve as a reference for the Hurwitz Zeta function, Lerch Transcedental, Polylogarithms and the Dirichlet Beta function which knowledge will be used in future chapters to solve actual problems (series, integrals and products). I´ve tried to use as elementary tools as possible, but in some instances some complex analysis, in particular contour integration was unavoidable. Hope you like it, and if you can leave a comment, suggestion of improvements and typos. Click here to access.

Today we will prove this beauty: Click here to see the proof.

In this post we will evaluate the follwoing pair of integrals. Click here to the proof:

Last week I came across a video of prof. Michael Penn where he proves the following result. Where G is Catalan´s constant . He solved this integral by a smart substitution as we will see in this paper, and left a challenge to solve this integral by another method. After many attempts I finally came up with a solution in terms of Polylogarithms, which is much harder and less intuitive but enabled me to generalize the above integral, namely: Click here to see the proof. Click here to see another collection of integrals related to Catalan´s constant

In this entry I will prove the remarkable Jacobi triple product using one of the most elementary proofs due to great mathematician George Andrews. Additionally, it will be shown some product representations of Jacobi theta function and the beautiful Euler´s pentagonal theorem This note is an excerpt of my ongoing personal project of putting all the blog content with additional content in a book format, so if you like it or would like to leave a comment please let me know. Click here to the proof.

Today I came across this integral in Twitter , click here for my proof.

In this post we will prove the following integral Click here to see the proof. We will need the following results previously established proved here , and proved here

Today we will prove the following result seen in this Twitter post : First we need a Lemma: Lemma 1: (1) Proof: Corollary: If we let in (1) we obtain (2) Now, consider the following integral (3) Proof: Where in (*) we used the following result proved here Now if we differentiate (3) with respect to s and let we get where we used that (see here ) and