Two questions...
a) by 'gamma function' do you mean...
... or some else?...
c) which is the path in the complex plane along which you intend to integrate the 'gamma function'?...
Kind regards
It says here that the residue of the gamma function at the simple pole z=–n is If C is meant to be the circle of radius 2 centred at –2+i then the poles inside the circle are at –1, –2 and –3, and times the sum of the residues given by that formula is indeed .