Contour integral (${\int }_{C}f(z)dz$)

For a piecewise differentiable parametrized contour $z(t)$, the contour integral is $\int f(z(t))z^{\prime }(t)dt$. It depends on the orientation of the contour. For a holomorphic function, the Cauchy integral theorem allows a contour deformation through regions without singularities.