MY MATEX NOTES

   Today we will show the following result that appears in this post from @integralsbot Let Then: And         We then get: By the reflection formula Letting we obtain that By the functional equation of the Gamma function We obtain for instance that

In this post we obtain the following integrals We have previously proved the following result (1) Letting we obtain (2) From (2), we can easily obtain the following relations (3) Lemma 1 (4) Proof: From (4) and (3) we obtain In the same fashion we obtain Lemma 2: (5) From Lemma 2 and (3) we obtain: Similarly

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 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