Fourier series of Loggamma function - Part 2
This post is part 2 of this post , where we started to derive the fourier series of the Loggamma function. Today we are going to evaluate the coefficient and then, put all the pieces together to find the final expression. Evaluation of Consider Integrating by parts now observe the following with regards to the limit: therefore (1) Recall now the integral representation of the Digamma function (2) Plugging (2) on the right hand side of (1) (3) Lets now concentrate in the inner integral in blue and break it down in two integrals, namely Lest begin with (4) Now This integral was already evaluated here , therefore (5) Plugging (4) and (5) back in (3), we get (6) We will consider each of these integrals separately To compute we perform an integration by parts (7) Where in the last line we used the resul...