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## Main Question or Discussion Point

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?

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?