An oriented piecewise differentiable path in the complex plane, used as the domain of a contour integral. Its orientation controls the signs of residues in the residue theorem.
A piecewise smooth curve has a continuous function as a parametrization on a compact interval, with a finite partition into subintervals on each of which the parametrization is continuously differentiable. One-sided derivatives may differ at joining points, so polygonal contours are allowed. A contour integral is the sum of the integrals over those smooth pieces.
The orientation of a contour specifies the direction in which its parametrization is traversed. Reversing that direction negates its contour integral. For a simple closed curve in the complex plane, positive orientation of a contour is the direction keeping its bounded interior on the left.
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