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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.

GAUSS DIGAMMA THEOREM

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February 26, 2023

Today we will prove the famous Gauss digamma theorem, namely

$\begin{align*} \psi\left(\frac{p}{q}\right) &=-\gamma-\frac{\pi}{2} \cot\left( \frac{\pi p}{q}\right)-\ln\left(q\right)+\sum_{j=1}^{q-1}\cos\left(\frac{2 \pi j p}{q}\right)\ln \left(2\sin \left(\frac{\pi j}{q} \right)\right) \end{align*}$

The proof is based on Jensen´s famous proof. Click here for the proof.

Digamma function

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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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