MY MATEX NOTES

In today´s post we will prove the following result by contour integration Click here to read the post.

          Today we will prove this infinite series which involves the reciprocal of the central binomial coefficient squared, namely Click here for the proof.

In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.

In this entry we will prove the following result that appears in this Twitter post Click here for the proof. In the proof we used the previous result proved here .

In this blog entry we will evaluate the beautiful series below involving the square of the central binomial coefficient Click here to see the proof We used the previous result proved here

Today we will prove the following generating function that appears in this Twitter post Click here for the proof.

In this post we will prove the following integral Click here to see the proof. We will need the following results previously established proved here , and proved here

Today we will prove the following result seen in this Twitter post : First we need a Lemma: Lemma 1: (1) Proof: Corollary: If we let in (1) we obtain (2) Now, consider the following integral (3) Proof: Where in (*) we used the following result proved here Now if we differentiate (3) with respect to s and let we get where we used that (see here ) and

Today we will prove the following result that appears in this Twitter post Click here to see the proof.

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 .

          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: