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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Today we will prove the following result that appears in this Twitter post
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 :
Today we will evaluate the following infinite sum which corresponds to the derivative of Dirichlet eta function @ 1 As a bonus we will compute the following integral Lets first introduce a lemma: Lemma 1: (1) Proof: Claim: (2) If we let in (1) we get (3) Lets now recall the Euler Maclaurin Formula (proved here ) to estimate the last two sums above (4) Choosing and in (4), we get for the first sum : (5) And for the second sum (6) Recall also the integral representation of the Stiltjies constant (shown here ): (7) letting in (7) we obtain (8) Now, plugging (5) and (6) back in (3) and letting we obtain We can now use (2) to calculate the following integral Proof:
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