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:

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