In general, the residue of at is the residue of at . Now we have . Therefore what you are looking for is the residue of at . The denominator is analytic at and its value is ; so the solution will be where is the residue of at ... can you finish now?
Show that for , the residue of at is .
is the gamma function. The gamma function is meromorphic. It is defined in the right half-plane by for . There is also another representation of where the right-hand side is defined and meromorphic for with simple poles at . However, I still do not see how to prove this. I need help with this. Thank you.