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

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.

For $\varphi (t)=r{e}^{it}$, ${\varphi }^{\prime }(t)=ir{e}^{it}$ and $|{\varphi }^{\prime }(t)|=r$. The triangle inequality for an integral applied to the parametrized contour integral givesThis is the circular-contour instance of the estimation lemma.