 Solution

For a piecewise continuously differentiable contour $\gamma :[a,b]\to U$, the contour integral is defined byIf the contour has corners, integrate on its finitely many differentiable pieces and sum the results. The integral on the right is taken componentwise as an ordinary real-parameter integral of a complex-valued function. The factor ${\gamma }^{\prime }(t)$ encodes both the speed and orientation of the contour.