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More specifically:

Suppose that $f(z)$ is meromorphic on a disk $\{\lvert z\lvert $f(z)$ on the annulus $\{s-\epsilon<\lvert z\lvert $f(z)$ at its poles.

Since $f(z)$ is meromorphic, the number of poles it has must be finite. However, I don't know how to define the disks surrounding each pole and add them up... Any help much appreciated.

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