MY MATEX NOTES

          In this post we will derive the Laurent expansion for the Riemann Zeta Function Recall the first order Euler Maclaurin expansion for the riemann zeta function shown here (1) We can rewrite (1) as following: Now, define the function: (2)   is an analytic function, therefore, we can represent it by a series expansion around in the following form: (3) Lets calculate the coefficients of (3) by calculating the derivatives of using eq. (2) above. For the first derivative we have: In the point we obtain for we obtain for we obtain for we obtain If we keep this process further we may obtain the following general form : (4) For . Evaluated at we obtain: (5) But we have already proved in a previous post that the integral in (5) is precisely the Stiltjies constants ! (6) Plugging (6) in (5) we get (7) for . To find we need to calculate Which we have also already calculated here and it´s equal to the E...

In this post we will show that (1) In order to prove (1) we can use the Euler Maclaurin formula derived previously in this post . The first order Euler Maclaurin Formula is given by: (2) Where , is the first Bernoulli Polynomial. Letting   and   , in (2) we obtain (3) (4) Lets now use (4) to prove (1) Proof:

          Today we will proof the following integral representation of the Euler Mascheroni constant that appears in this post We will first proof the following Lemma Lemma: Proof: Consider the following complex function and lets integrate it around the contour C below. Clearly from Cauchy´s theorem the integral is equal to zero Now, Taking limits Equating Real and Imaginary parts we obtain (1) From (1) we conclude that (2) Letting in (1), then (3) From (2) and (3) it follows that (4) Lets now compute the R.H.S. of (4) Where in the last line we used the result proved here . Therefore we conclude that Or