MY MATEX NOTES

Today we will prove the following result seen in this Twitter post : First we need a Lemma: Lemma 1: (1) Proof: Corollary: If we let in (1) we obtain (2) Now, consider the following integral (3) Proof: Where in (*) we used the following result proved here Now if we differentiate (3) with respect to s and let we get where we used that (see here ) and

In this short post we will estabilish the following result Where in (**) we used the previous established results ( I and II ): In (*) above we used the following result

In today´s post We will compute these wonderfull integrals: From last post we know (1) (2) (3) Lemma 1 Proof: Therefore, from Lemma 1 and from (1) and (2) we obtain Lemma 2 Therefore from Lemma 2 and (2) and (3) we obtain

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