# Residues of the Gamma function

I am trying to make sense of a proof that the poles of $$\Gamma(z)$$ are at $$z=-n$$ and have residue $$\frac{(-1)^n}{n}$$. The proof reduces $$\Gamma(z)$$ to the sum of an (entire) incomplete gamma function and a meromorphic part given by $$\sum_{n=0}^\infty\frac{(-1)^n}{n!(z+n)}$$ My issue is the expansion $$\frac{1}{z}-\frac{1}{1+z}+\frac{1}{2(z+2)}- ...$$ does not resemble a Laurent Series since each term is centered around a different point. Can someone explain why it is still valid to take the coefficient of $$(z-z_0)^{-1}$$ in this form in order to find the residue at $$z=z_0$$?

• Where did you find this proof? – José Carlos Santos Oct 26 '18 at 10:49
• Complex Analysis lecture slides – P Collier Oct 26 '18 at 10:52
• – Nosrati Oct 26 '18 at 11:09
• Oh yes I see now that you can simply recombine the terms without worrying about a Laurent series, thanks for pointing me in the right direction! – P Collier Oct 26 '18 at 11:17