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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.

A simple proof of Lerch´s formula

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March 24, 2022

In this blog entry we will present a simple proof for Lerch´s formula

$\ln\left(\Gamma\left(x \right) \right)=\zeta^\prime(0,x)-\zeta^\prime(0)$ (1)

We rely on the previous established result (see proof here)

$\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{x + t} dt =\left(x-\frac12\right)\ln x-x+\frac{\ln (2\pi)}{2}-\ln\left( \Gamma (x)\right)$

Click here to see the proof of (1) .

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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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