MY MATEX NOTES

         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

I came across this beautiful integral today in this twitter . Lets evaluate it Lets focus in the inner integral Then Putting back the values of J and K we obtain

          Today we will proof the following integral representation of the Euler Mascheroni constant that appears in this post We will first proof the following Lemma Lemma: Proof: Consider the following complex function and lets integrate it around the contour C below. Clearly from Cauchy´s theorem the integral is equal to zero Now, Taking limits Equating Real and Imaginary parts we obtain (1) From (1) we conclude that (2) Letting in (1), then (3) From (2) and (3) it follows that (4) Lets now compute the R.H.S. of (4) Where in the last line we used the result proved here . Therefore we conclude that Or

Today we will evaluate the following integral found in this post Lets consider the more general case Now observe the following: First recall the Beta function Differentiating the above expression w.r. to a we obtain Letting we obtain Now let   We used the reflection formula of the Gamma function and the special values of the Digamma function at