This is addressed to people who know complex analysis (hope this is the right section). Here's the Laurent theorem from my book for my later reference: Suppose a function f is analytic throughout an annular domain R1<|z-z0|<R2, centered at z0, and let C denote any positively oriented simple closed contour around z0 and lying in that domain. Then, at each point in the domain, f(z) has the series representation f(z)=sum (n=0 to inf) of an(z-z0) + sum(n=1 to inf) bn/(z-z0)^n, where an=(1/ 2pi*i) integral over C of f(z)dz/(z-z0)^(n+1) and bn=(1/ 2pi i) integral over C of f(z)dz/(z-z0)^(-n+1). So say there is a finite number of singularities of f, so they're all in a circle of some radius R. It seems to me you could select any point p in the complex plane, and then make an annulus around that point such that its inner perimeter encloses the singularities. The theorem seems to say that 2pi i *Res(f,p) is equal to the integral of a closed contour around the singularities. I know this is wrong b/c it doesn't agree with what I've seen with the Cauchy Residue Thm, where I calculated that integral by summing residues at singularities. Where is the disconnect?