MY MATEX NOTES

Let´s proof today the following integrals Recall the integral representation of the Gamma Function We can differentiate under the integral sign with respect to to obtain the following results: setting we obtain (1) (2) (3) (4) On the other hand, we can obtain expressions for the derivatives of the Logarithm of the Gamma function as following (5) (6) (7) (8) Now recall the special values of the polygamma function (9) (10) (11) (12) Setting in (5) and using (9) we obtain (13) Equating (1) and (10) we get (14) Setting in (6) (15) Equating (2) and (15) we get (16) Setting in (8) (17) Equating (3) and (17) we get (18) Setting in (7) (19) Equating (4) and (19) we get (20)

      Following the previous two posts 1 , 2 , today we will compute the following two infinite sums, probably the last post about these sort of sums.       As before, the first sum is pretty straightforward, but the second one is much harder and demanded a good amount of computations. The first sum I could find the close form result to check our answer. The second one I could not find anywhere, but checked Wolfram Infinite sum calculator and it matches numerically! The values of   can be found here , then Plugging the results (A.6),(A.9),(A.8) and (A.4) in the above equation we obtain And finally! Appendix Lets start computing the following indefinite integral Let , then Let , then (A.1) Let , then Let , then (A.2) (A.3) Plugging (A.1) and (A.2) in (A.3) Applying formula (A.11) we obtain (A.4) Note that   (A.5) Plugging (A.1) and (A.2) in (A.5) we obtain (A.6) Recall the formula (A.7) (A.8) Ap