Editing Contour integration (section)

===Directed smooth curves===
Contours are often defined in terms of directed smooth curves.<ref name=Saff/> These provide a precise definition of a "piece" of a smooth curve, of which a contour is made.

A '''smooth curve''' is a curve <math>z:[a,b]\to\C</math> with a non-vanishing, continuous derivative such that each point is traversed only once ({{mvar|z}} is one-to-one), with the possible exception of a curve such that the endpoints match (<math>z(a)=z(b)</math>). In the case where the endpoints match, the curve is called closed, and the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point (<math>z'(a)=z'(b)</math>). A smooth curve that is not closed is often referred to as a smooth arc.<ref name=Saff/>

The [[Parametrization (geometry)|parametrization]] of a curve provides a natural ordering of points on the curve: <math>z(x)</math> comes before <math>z(y)</math> if <math>x<y</math>. This leads to the notion of a '''directed smooth curve'''. It is most useful to consider curves independent of the specific parametrization. This can be done by considering [[equivalence classes]] of smooth curves with the same direction. A '''directed smooth curve''' can then be defined as an ordered set of points in the complex plane that is the image of some smooth curve in their natural order (according to the parametrization). Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. Also, a single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints.