GAMMA REFLECTION FORMULA VIA CONTOUR INTEGRATION
We have already proved this famous result in a previous post via real methods. Today I want to proof it via contour integration and in the end of the post, use it to compute a famous infinite series. We start by considering the following integral . Lets evaluate the complex integral where It´s clear that has a pole @ and a branch @ . We choose to be the keyhole contour below. We can Write (1) Let´s first calculate the residues (2) Now let´s focus in each integral. Lets start by the integral around the big Arc and show that it vanishes as letting . We have: Then Since Similarly, letting over we get Therefore, from (1) and (2) we have (3) Lets start by the noting the following fact Than now let in the inner integral From the standard integral representation of the Gamma function (4) Equating (3) and (4) obtain the reflection formula for the Gamma function: (5) if we enforce (6) If we take...