MY MATEX NOTES

Today´s blog we will evaluate a hard integral, namely: Where is Catalan´s constant (G=0.915965594177219015054603514932384110774)  and is the Polylogarithm function . Recall the expansion (see here ) (1) Therefore (2) Integrating both sides of from 0 to We used that (see Appendix below): And Appendix Recall the generator function (see this post) (A.1) Letting in (A.1) we obtain We are looking for the Imaginary part of the equation above, hence Where we used the following results(see proofs here ): Recall (see this post ) (A.2) Integrating twice both sides of (A.2) from 0 to x we obtain: Letting we obtain:

Today´s post We will compute the following logsine integral Analogous to what we did previously in this post with regards to the pair We will find a series representation to and then use it to the evaluation. Recall the following relations: (1) and (2) and by Euler´s formula we have (3) But the left hand of (3) is equal to (4) And therefore we conclude that (5) If we square both sides of (5) we obtain (6) Which gives us (7) (8) Now recall the generating function of Harmonic numbers ( see here ) (9) integrating both sides of (9) w.r. to x (10) (11) Now if we let in (11) we obtain (12) Which gives us (13) (14) Comparing equation (7) with (13) we conclude that (15) Or (16) Now, lets evaluate the integral Where we used the results proved here . Also for the integral: Check here the entry where we computed the sister of today´s integral:

In today´s post we will compute the amazing alternate infinite sum Where in (*) we used the generating function previously proved here Evaluation of   Evaluation of Where we used the following result Evaluation of Where We used the previously proved results Evaluation of Appendix Proof: We start from On the other hand Equating both equations we obtain the desired result