Clausen Function, Logsine Integrals and Central Binomial Series
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 od...