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.
Inverse hyperbolic tangent integral
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In this entry we prove the follwoing result, an integral I came across today in this video
Today We will compute the following integral following the same ideas of the previous post : Recall (see here ) (1) Integrating both sides of (1) from 0 to Computing Recall (see here ) Letting we obtain We are looking for the Imaginary part of the equation above: Computing the quantities: The Glaisher function We know that (see here ): If we integrate from 0 to x we obtain Integrating from 0 to x we obtain
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.
Four integrals evaluated with Euler´s formula trick In this post I intend to evaluate four basic integrals that will turn to be very useful in future posts for evaluating more complex and interesting integrals. Usually these integrals are evaluated integrating by parts twice, which renders a quite tedious and long task. The idea here is to use Euler´s formula , namely: to turn these integrals into simple exponential integrals which are very easy and straightforward to evaluate and consequently each pair of integrals will be evaluated through only one integral. So lets get started! The first pair of integrals that I want to consider are the Laplace transform of sin(at) and cos(at). (1) (2) (3) And we get that: Note that and which gives us that: The next pair of integrals looks very much like the previous one with except that now the limits of integration are from zero to one instead of zero to infinit...
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