MY MATEX NOTES

          Today we will prove this infinite series which involves the reciprocal of the central binomial coefficient squared, namely Click here for the proof.

In this blog entry we will prove the following result: Click here for the proof.

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

Last week I came across a video of prof. Michael Penn where he proves the following result. Where G is Catalan´s constant . He solved this integral by a smart substitution as we will see in this paper, and left a challenge to solve this integral by another method. After many attempts I finally came up with a solution in terms of Polylogarithms, which is much harder and less intuitive but enabled me to generalize the above integral, namely: Click here to see the proof. Click here to see another collection of integrals related to Catalan´s constant

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:

Let´s proof the following two infinite sums related to Catalan´s constant Where is the Catalan´s constant Recall (see here ) (1) Dividing both sides by x and integrating from 0 to 1 (2) Than Dividing both sides of (1) by x and integrating from 0 to 1/2 we obtain (3) Than Where we used the result proved here :

               In this post We will proof the following integrals related to the Catalan´s Constant . The proofs are based on the following paper: Representations of Catalan’s constant, David Bradley, 2001 (1) Proof: (2) Proof: (3) Proof: (4) Proof: By equation (A.7) of the appendix we have Than, we can rewrite our itegral as Equating (3) and (4) we conclude that (5) Letting in (5) we obtain (6) We can Integrate by part (6) to get And from (6) we conclude that (7) Now, let in (7) Ans from (7) we conclude that (8) Appendix (A.1) Proof: Recall the addition formulas of sine and cosine (see here ) (A.2) (A.3) Examples: Another two examples that will be useful in the next proof (A.4) Proof: Recall the double angle formulas (see here ): (A.5) (A.6) Than (A.7) Proof: Reference Representations of Catalan’s constant, David Bradley, 2001