Contour

In complex analysis, a contour $γ : [a, b] \to C$ is a piecewise smooth, oriented curve in the complex plane, typically used as a path of integration in complex integration. Piecewise smooth means that there exists a partition

a = t_{0} < t_{1} < \dots < t_{n} = b

such that on each subinterval $[t_{k}, t_{k + 1}]$ , $γ$ is continuously differentiable.

When we say a contour is closed, it means $γ (a) = γ (b)$ .
A simple closed contour does not cross itself and encloses a region without overlapping.