MY MATEX NOTES

         In today´s post we will review the Clausen function and prove some of its properties. Than we will show it´s connection with  logsine integrals and Binomial series. The Clausen function is defined as Recall the Fourier expansion (see here ) (1) Integrating both sides from 0 to x we obtain (2) Claim 2: (3) Proof: On the other hand Hence Sometimes it´s useful to integrate the Clausen function: (4) Proof: (5) Proof: From equation (2) we can derive a duplication formula for (6) Proof: the same procedure we can obtain a duplication formula for (7) Proof: Integrating both sides of (6) form 0 to x we have: Lets evaluate each of these integrals separately Putting all together The general formula is given by: (8) First note that (very easy to prove just by expanding the R.H.S.): (9) Then (10) On the other hand (11) By (9) we can equate (10) and (11) (12) Proving the duplication formula for odd indices. For even indices we di

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: