An alternate infinite series involving sinh(n \pi)
In this entry we present a proof via contour integration for the following alternate infinite series Click here for the proof.
Posts
Two remarkable sums due to Ramanujan-Part II
In this entry we present a second proof via contour integration for the series (proved previously here ) Click here for the proof.
Two remarkable sums due to Ramanujan-Part I
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.
Another quick contour integral from the integralbot
In this blog entry we solve another integral from @integralbot via contour integration. Click here for the proof.
Transformation formula Dedekind´s eta function
In this blog entry we prove the transformation formula for the Dedekind´s eta function The proof is due to C.L. Siegel, and it´s done by contour integration. Click here to read it.
A Ramanujan Series
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
Integral of an infinite product
In this article our goal it to prove the following integral To this end we will rely on two results: Euler´s pentagonal theorem which states that we have proved this before ( see here ). The second tool that we will use regards the evaluation of alternating infinite series using the residues theorem to evaluate the following infinite series Click here for the proof .
QUICK CONTOUR INTEGRAL FROM @integralbot
In today´s post we will prove the following result by contour integration Click here to read the post.
Elliptic integrals and the arithmetic-geometric mean
In this entry we introduce the Arithmetic Geometric mean and show it´s connection with the complete elliptic integral of the first kind. This relation is one way to establish the connection between the Jacobi theta function and elliptic integral. and Click here to read the article. Important to note that in this entry we wont get into a detailed study of elliptic integral. Commentaries and feedbacks are welcome.
Ramanujan´s Psi Sum
In this blog entry we will prove Ramanujan´s Psi Sum, namely Click here for the proof. As corollaries of the above formula we show that wich relies also on Jacobi triple product proved here :
A log trig integral \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx
In this post we will prove the following result Click here for the proof We have relied on the previous established result Proof here .
Trig integral equals alternate inverse reciprocal binomial series
In this entry we will solve the following integral to this end we should recall a previous established result. Click here for the proof of the integral. Proof of the binomial series here .
log log integral inverse sech
In this entry we will prove the following result Click here to read it. Some previous established results were used, i.e.: Lerch´s formula Click here for the proof. Relation between Bernoulli´s polynomials and Hurwitz zeta function click here for the proof (equations 8.22 and 8.30)
INFINITE PRODUCTS AND JACOBI THETA FUNCTION
This entry has some of the infinite product related to Jacobi Theta function proved. Click here to read it. The Jacobi triple product proved here , plays a central role in their proof as well some special values of the Jacobi theta function, see here
Series involving reciprocal of the central binomial coefficient squared
Today we will prove this infinite series which involves the reciprocal of the central binomial coefficient squared, namely Click here for the proof.
Central Binomial representation for zeta(2)
In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.
Central Binomial coefficient series and Catalan Constant
In this blog entry we will prove the following result: Click here for the proof.
Inverse hyperbolic tangent integral
In this entry we prove the follwoing result, an integral I came across today in this video Click here for the proof.
Central Binomial coefficient series and zeta(2)
In this entry we will prove the following result that appears in this Twitter post Click here for the proof. In the proof we used the previous result proved here .
Series involving central binomial coefficient squared
In this blog entry we will evaluate the beautiful series below involving the square of the central binomial coefficient Click here to see the proof We used the previous result proved here
Central Binomial coefficient generating function
Today we will prove the following generating function that appears in this Twitter post Click here for the proof.
CHAPTERS HURWITZ ZETA, LERCH, POLYLOGS AND DIRICHLET BETA
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.
INTEGRAL INVOLVING ARCTAN SQUARED PART -1
Today we will prove this beauty: Click here to see the proof.
PAIR OF ARCTAN INTEGRALS
In this post we will evaluate the follwoing pair of integrals. Click here to the proof:
MICHAEL PENN´S CHALLENGE INTEGRAL
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
JACOBI TRIPLE PRODUCT AND PRODUCT REPRESENTATION OF JACOBI THETA FUNCTIONS
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.
CHALLENGING INTEGRAL INTEGRALBOT
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
HYPERBOLIC INTEGRAL WITH GLAISHER CONSTANT
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