Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 3 / a / i

For a piecewise continuously differentiable contour $\gamma :[a,b]\to U$, define its contour integral byadding 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.