MY MATEX NOTES

In this post we will compute the following three integrals: Consider the following three integrals (1) (2) (3) Differentiating the three w.r. to z we obtain (4) (5) (6) Now recall Binet´s Integral representation for the Digamma function (7) Making the change of variable and multiplying (7) by 4z we get (8) Multiplying (3) by (9) Integrating (8) w.r. to z   Letting Where we used that Proved here . For the second integral, integrating from 0 to z Letting Where we used that Proved here . For the last integral, We Multiply both sides of (7) by to get Now let And now let to obtain Integrating from 0 to z Letting

In this post we will proof the following two integrals related to the derivative of the Riemann Zeta function at Recall Kummer´s fourier expansion for LogGamma    (1) Integrating (1) from 0 to (2) Letting in (2) Where we used that  Proved here , and that For the second integral, lets first evalute the following integral                                                Then, differentiating both sides w.r. to s Setting s=2 Setting Where we again made use of the fact that

In this short post we will evaluate the following integral Where in (*) we used the previously established result In (**) we made use of the Lemmas in the appendix below. (1) Letting and in (1) we obtain (2) Letting and in (1) we obtain (3) Lemma 1 Lemma 2 Proof: We have (A.1) (A.2) Multiplying (A.1) by (A.2) we obtain Lemma 3 Proof: We have (A.3) (A.4) Multiplying (A.3) by (A.4) we obtain Lemma 4 Proof: We have Multiplying (A.2) by (A.3) we obtain