MY MATEX NOTES

In continuation of our previous post about Malmsten integrals , we will today prove the following four integrals: For the First Integral Where We used Vardi´s integral: and that To evaluate the second integral, we first consider the following integral If we differentiate the above result w.r. to we obtain Letting Then to compute our integral Where we used that (1) Differentiating     w.r. to (2) We have also previously proved here that (3) and (4) Setting in (1) and (2) we get For the third integral, recall the previous established result If we let   , then For the last integral let in Then Where we used the reflection formula To write

We will today prove the following integral that belongs to a family of log-log integrals known as Malmsten integrals  which Vardi´s integral is a particular case: Mamlsten Integrals (1) If we let      in (1) we obtain the desired result: (2) Appendix: Cauchy Product Example We have that and , we therefore obtain Evaluation of the integral: Recall the sine of a difference formula And The Fourier series for the LogGamma function proved here If we let       We obtain Or

         Today´s post we will evaluate the following three interrelated integrals Dirichlet Beta function We should first recall Dirichlet´s Beta function Then, if we differentiate it w.r. to we obtain Letting we obtain (1) Now, recall Kummer´s expansion for the LogGamma function : (2) Rearranging terms we get (3) If we let        in    we obtain Or (4) Comparing (4) with (1) we conclude that (5) Now let´s evaluate the first integral, namely: (6) Differentiating (6) w.r. to we get (7) Letting in (7) we obtain Two related integrals. The first one is known as Vardi´s integral