Search This Blog

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

Get link
Facebook
X
Pinterest
Email
Other Apps

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) .

Get link
Facebook
X
Pinterest
Email
Other Apps

Comments

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.

Labels

@infseriesbot
@integralsbot
AGM
Apery´s constant
Beta function
BOOK
Catalan´s constant
cauchy-schlomilch
Central Binomial
Clausen Function

contour integral
convolution
Dedekind´s eta function
Digamma function
dilogarithm
Dirichlet Beta function
Dirichlet Eta function
double sum
Eisenstein series
elliptic integrals
Euler Maclaurin
Euler Sums
Euler-Mascheroni
Euler´s equation
Fourier Series
Fourier Transform
Gamma function
Gauss Multiplication Formula
Glaisher Kinkelin
golden ratio
Gradshteyn and Ryzhik
Harmonic numbers
Hurwitz Zeta function
indefinite integral
Infinite Product
Infinite Series
integral representation
Integrals
jacobi theta function
Jacobi triple product
Kummer
Laplace Transform
Laurent expansion
Loggamma
logsine
logsinh
Mamelsten Integrals
Mellin Transform
Modified Bessel Function
partial fractions
Poisson Summation
Polygamma function
polylogarithms
polylogathm
probability
Ramanujan
Reflection Formula
residues theorem
Riemann Zeta
roots of unity
special functions
Stiltjies constants
Trigamma function
Vardi´s integral
Weiestrass Infinite Product
Zeta Function

Show more Show less