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Problem 2

Let \(f(z)\) be analytic inside and on a simple closed contour \(C\) except for a finite number of poles inside \(C .\) Denote the zeros by \(z_{1}, \ldots, z_{n}\) (none of which lies on \(C\) ) and the poles by \(w_{1}, \ldots, w_{m} .\) If \(g(z)\) is analytic inside and on \(C\), prove that $$ \frac{1}{2 \pi i} \int_{C} g(z) \frac{f^{\prime}(z)}{f(z)} d z=\sum_{j=1}^{n} g\left(z_{j}\right)-\sum_{j=1}^{m} g\left(w_{j}\right) $$ where each zero and pole occurs as often in the sum as is required by its multiplicity.

Expert verified

The given equation is proven using the residue theorem and the properties of a contour integral in complex analysis. The crucial part in this proof is calculating the residues at the zeros and poles of \( f(z) \).

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