MY MATEX NOTES

Today we will prove the famous Binet´s formulas for Log Gamma function, namely: Let´s start by computing the integral Where we used the result proved here Then,                            (1)                               Now recall Stirling’s approximation for the Gamma function (2) Taking logarithms in both sides of (2) (3) Plugging (3) in (1) and taking the limit The L.H.S. goes to zero, and we conclude that                                                             Therefore we get Now for the second Binet´s relation, consider the Integral Where we have used in the second line. Now make the following substitution,   to get: (4) Differentiating (4) w.r. to z This last integral we have already computed here , it´s value is Then (5) Following the same procedure as before we obtain

Today we will proof the following two infinte sums found here First, lets evaluate the following infinite sum: Recall the following result (1) For we obtain Lets now differentiate (1) w.r. to a (2) Letting in (2) Now, for the second sum

Today, we will proof the following relations: (1) (2) (3) (4) Recall the infinite sine product (5) To proof (2), we recall the relation , so letting in (1) letting       in (1) we obtain Similarly, letting in (2) we obtain Corollary Taking log in both sides of (2) From the relation   

Today, we will proof the result that appears here , namely: Lets start by evaluating an easier infinte sum that is related to our goal Sum. (1) If we let in (1) we get Applying (A.7) and (A.8) we obtain (2) Now, observe the following Using result (2) (3) Appendix (A.1) Proof: Recall (A.2) (A.3) (A.2) - (A.3) (A.4) Proof: Recall (A.5) (A.6) Then, (A.5) - (A.6) (A.7)                           (A.8) Proof: