A log trig integral \int_0^{\pi/2}\ln\left(1+a\sin^2 x \right) \,dx
In this post we will prove the following result Click here for the proof We have relied on the previous established result Proof here .
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Series involving reciprocal of the central binomial coefficient squared
Today we will prove this infinite series which involves the reciprocal of the central binomial coefficient squared, namely Click here for the proof.
Central Binomial representation for zeta(2)
In this post we will prove the following nice series representation for : Click here for the proof. We used the previous established result Click here for the proof.
Central Binomial coefficient series and Catalan Constant
In this blog entry we will prove the following result: Click here for the proof.
Central Binomial coefficient series and zeta(2)
In this entry we will prove the following result that appears in this Twitter post Click here for the proof. In the proof we used the previous result proved here .
Series involving central binomial coefficient squared
In this blog entry we will evaluate the beautiful series below involving the square of the central binomial coefficient Click here to see the proof We used the previous result proved here
Central Binomial coefficient generating function
Today we will prove the following generating function that appears in this Twitter post Click here for the proof.
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...