 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-4/4a/solution

For a closed piecewise continuously differentiable path avoiding

a

n (γ, a) = \frac{1}{2 πi} \oint_{γ} \frac{d z}{z - a} = \frac{1}{2 πi} \int_{0}^{1} \frac{γ ^{'} ( t )}{γ ( t ) - a} d t .

Break the parameter interval at the finitely many derivative discontinuities when evaluating this integral. Translating the curve gives

n (γ, a) = n (γ - a, 0)

, which will also be used below.

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