An easy looking integral, not so easy...PART 2
In this previous post We have computed the following integral (1) Today we are going to compute the third integral of the list, namely: Lets get to it! Start with By (1) we have (2) We now focus in Integrating by parts (3) So now, the task is to evaluate the Sum (3) To calculate this sum, consider the following generating function proved here eq. (10) Letting The left hand side becomes We therefore have that or Lets now compute the components of (4) and then, put all together to get the solution. (5) (6) and (7) (8) Recall the Dilogarithm reflection formula ( proof here , eq. (4)) Plug (9) (10) Recall the identity proved here equation (5) therefore but and (12) on the other hand (13) From (12) and (13) we conclude that (14) We can now rewrite (11) as (15) From (4) and (6) we have (16) (17) Plugging the real part of (10), (15), (16) and (17) in (4) we get Consequently we get t