TITLE: Calculus of residue of function around poles of fractional order (complex analysis)
QUESTION [11 upvotes]: The complex function $f(z)=\frac{1}{\sqrt{z^2+r_0z}}$ with $r_0>0$ has two poles (at $z=0$ and $z=-r_0$). But they are not simple poles. They are poles of fractional order. Am I right? How I can calculate residue of the function at the poles? Please help me.

Thanks
Vahid

REPLY [10 votes]: The answer is that these are not poles - they are branch point singularities.  They are not covered by the residue theorem, and if you were to include this function in a contour integral, you would want to point your contour so as to not include the branch cut $z \in [-r_0,0]$ inside the contour.