A parametric representation of a contour C means a continuous function from an interval on the real line to C, such that as t goes along the interval, s(t) traces out the contour C.

For example, if C is the unit circle, you can parametrise it by the map for t in the interval . As the parameter t goes from 0 to 2π, s(t) goes right round the contour C.

If C is the contour given by y = x^3 (where z=x+iy) then you can take x to be the parameter. The parametrisation is then the map , defined on the interval [–1,1]. As x goes from –1 to 1, s(x) goes along the curve from –1–i to 1+i.

If you want to integrate a function f(z) along the contour C, then you substitute z=s(x), dz=s'(x)dx, and the integral becomes an integral of a (complex-valued) function of a real variable, which you can integrate using any of the techniques of ordinary (real-variable) integration. So

(we have to split up the interval of integration because the function f is defined differently depending on whether y (=x^3) is positive or negative)

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