MY MATEX NOTES

In this blog entry we will present a simple proof for Lerch´s formula (1) We rely on the previous established result (see proof here ) Click here to see the proof of   .

In this post we will present proofs for the follwoing two integrals related to the LoGamma function: Click here to see it.

We want to prove the following transformation formula that appears in this twitter post Click here to see the proof . We used the result proved here .

Today we will show the following result that appears in this twitter post We will start by proving two lemmas involving the Beta and Digamma functions. Lemma 1: (1) Recall the Beta function (2) If we differentiate (2) w.r. to s we obtain Lemma 2: (3) Recall the Beta function (4) If we differentiate (4) w.r. to a we obtain Now let´s evaluate the integral: For the proof of the special values of the Digamma function see this post . Plugging the values of J and K back in the original integral we obtain:

          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:

          Let´s prove today the following beautiful  relations of the derivative of the Riemann Zeta function: We showed previously ( here ) the functional equation of the Hurwitz Zeta function (1) If we set h=k=1 in (1) we obtain (2) Which is the functional equation of the Riemann zeta function. Let´s now differentiate equation (2) w.r. to s to obtain: (3) Letting , n a positive integer and noting that   and     we may obtain (4) Now plugging in (4) we obtain the desired relations:

          In today´s post we will derive a functional equation for the Hurwitz Zeta Function, namely: To this end we introduce the periodic zeta function defined by the following expression: (1) Theorem 1: (2) for and Proof: First recall the previously proved fourier expansion (see here ) (3) for and Now lets expand (2) by means of Euler´s formula Lemmma 1: Let h and k be two integers, , then for (4) Proof: To prove this equality used in the proof above Lets expand first it´s L.H.S. We used the fact that for m an integer Now, lets expand it´s R.H.S. and show that they are equal to each other Theorem 2: If k and h are integers with , then for all s we have Proof: Plugging (4) in (2) we obtain: