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Prove: Let f(z) be analytic inside and on a simple, closed curve C , except for j poles of orders
m_{1}, m_{2}, ... m_{j} at points α_{1}, α_{2}, ..., α_{j} and k zeros of orders n_{1}, n_{2}, ... n_{k} at points β_{1}, β_{2}, ..., β_{k}, all
located in the region inside C. Suppose that f(z)
0 on C. Then

Proof. Enclose each of the poles by non-overlapping circles Γ_{1}, Γ_{2}, ... , Γ_{j} and each of the
zeros by non-overlapping circles Σ_{1}, Σ_{2}, ... , Σ_{k , }all contained within C, as shown in Fig. 1. Then

= (- 2πim_{1} - 2πim_{2} - ... -2πim_{j }) + (2πin_{1} + 2πin_{2} + ... + 2πin_{k})

= 2πi [(n_{1} + n_{2} + ... + n_{k}) - (m_{1} + m_{2} + ... + m_{j})]

or

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