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Prove: Residue theorem. Let f(z) be
analytic inside and on a simple closed curve C
except at the singularities a, b, c, ... inside C
which have residues given by a_{r}, b_{r}, c_{r} ... .
Then

Proof. Construct circles C_{1}, C_{2}, C_{3}, ...
centered at a, b, c, ... , of such radii as to lie
entirely within C as shown in Fig. 1.

Now by Theorem 6 in the section on Complex integration and Cauchy’s theorem we have

Since

we have

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