 Solution

ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2017/3/a/i/solution

For a piecewise continuously differentiable contour

γ : [a, b] \to U

\int_{γ} f (z) d z = \int_{a}^{b} f (γ (t)) γ^{'} (t) d t,

adding the integrals over the finitely many smooth pieces if necessary. Continuity of

f

along the contour makes each integrand integrable. The derivative supplies both the direction and the complex line element; replacing it by its modulus would give a different integral with respect to arc length.

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