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MY MATEX NOTES

This is a collection of personal notes from a non mathematician with techniques that I have learned throughout the years focused mostly in integrals. It´s mainly goal is to organize my manuscripts in a more cleaner and readable format and share with whoever finds it interesting.

ELLIPTIC INTEGRAL HYPERBOLIC FUNCTIONS

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May 12, 2022

Today we will prove the following result that appears in this Twitter post

$\begin{align*} \int_0^{\pi/2}\frac{d \theta}{\left(\frac13 + \sin^2(\theta) \right)^{1/3}}&=\frac{\sqrt[3]{3}}{\sqrt[3]{2^7}}\cdot\frac{\Gamma^3\left(\frac13\right)}{\pi} \end{align*}$

Click here to see the proof.

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Ramanujan´s Psi Sum

September 29, 2022

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 :

Infinite sum reciprocal hyperbolic sine squared

January 18, 2023

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.

LAPLACE TRANSFORM OF A COMPLEX POWER

August 17, 2023

In this post we will show that For , where and . As a Corollary we also obtain the beatiful pair of integrals Click here for the proof.

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