MY MATEX NOTES

        Today we will evaluate the beautiful integral found in this post We have (see appendix bellow for the proof) Then Appendix Recall the generating function Lets now focus only in the integral of the R.H.S. We then conclude that (A.1) Recall (A.2) (A.3) (A.4) On the other hand (A.5) Therefore (A.6) Squaring both sides of (A.6) we obtain (A.7) Now, let in (A.1), we obtain (A.8) Plugging the R.H.S. of (A.7) in the R.H.S. of (A.8) we get (A.9) Equating Real and Imaginary parts in (A.9) we obtain (A.10) (A.11) (A.11) Recall that      ,     then Lets integrate by part the remaining integral, considering that       Integrating by parts     Therefore (A.12)

               Today I came across the following relation in this Twitter post by @infseriesbot . We will show today how to proof this Interesting relations. Consider the following three Dilogarithm relations (1) (2) (3) (1) is proved in the end of the post (equation (A.5)). The proof for (2) and (3) can be found in this previous post . The first relation can be proved automatically just by letting          in     (2) For the other three relations, recall the Golden ratio it´s easy to verify that letting        in (1), (2) and (3) we obtain (4) (5) (6) Subtracting (6) from (4) we get (7) Now Subtracting (7) - (5) gives which gives us (8) Plugging (8) in (5) gives (9) Plugging (8) in (6) gives (10) Appendix Recall the following representation of the Gamma function let (A.1) Let and (A.2) now consider the definition of polylogarithm...