MY MATEX NOTES

In this post We will prove the following integral Lets first evaluate the integral (1) Differentiating (1) w.r. to s Now, letting (2) Now recall that (3) And consequently (4) We also have the following relations ( proved here ) and Setting in (3) and (4) we get (5) (6) Plugging (5) and (6) in (2) Appendix Recall the geometric series

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

Some fun integrals for Friday: We used (1) (2) (3)

Today we will prove the following two amazing integrals (1) (2) In order to prove (1), first recall the following results: (3) (4) Then, (5) If we let and in (5), we get (6) Where We used the result We can rewrite (6) as (7) If we now integrate (7) w.r. to z we have: The evaluation of the constant is a beautiful exercise per se. Fortunately relying on the previous estabilished Vardi´s integral We may accomplish it easily. Setting in the last equation, the L.H.S. becomes Where we used the Vardi´s integral proved here : And The R.H.S. becomes Equating L.H.S. and R.H.S. we conclude that And finally (8) Or (9) Appendix Recall Legendre Duplication Formula for the Gamma Function Letting

Following the previous post , We will evaluate the following two integrals today Recall Kummer´s fourier expansion for LogGamma    (1) Multiplying (1) by and integrating form 0 to 1 where we used Multiplying (1) by and integrating form 0 to 1 Appendix

                     In this blog entry we will derive the functional equation for the Riemann Zeta Function. It extends the Zeta function to the entire complex plane except for the point which is a simple pole. We have already found an analytic continuation for the Zeta function through the Euler Maclaurin summation formula . There, we were able to extend it´s domain to the left side of the complex plane step by step increasing the order of the Euler Maclaurin formula. The functional equation enables us to extend the Zeta function to the entire complex domain at once. We will start by first introducing the Poisson summation formula which is a key ingredient in the derivation. In the end of the post we will show one small branch of it´s applicability proving the result   , which we have extensively used computing integrals. Poisson Summation Formula Let     be a continuous function of defined for Let is periodic with period Proof: Letting we obtain Then, can be expand