Editing Contour integration (section)

==Curves in the complex plane==
In [[complex analysis]], a '''contour''' is a type of curve in the [[complex plane]]. In contour integration, contours provide a precise definition of the [[curve]]s on which an integral may be suitably defined. A '''curve''' in the complex plane is defined as a [[continuous function]] from a [[closed interval]] of the [[real line]] to the complex plane: <math>z:[a,b]\to\C</math>.

This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down the set of curves that we can integrate to include only those that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, and we require that each piece have a finite (non-vanishing) continuous derivative. These requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of even, steady strokes, which stop only to start a new piece of the curve, all without picking up the pen.<ref name=Saff>{{Cite book|title=Fundamentals of Complex Analysis with Applications to Engineering, Science, and Mathematics |edition=3rd |chapter=Chapter 4 |url=https://books.google.com/books?id=fVsZAQAAIAAJ&q=saff+%26+Snider
|first1=Edward B. |last1=Saff |first2=Arthur David |last2=Snider |year=2003 |publisher=Prentice Hall |isbn=0-1390-7874-6}}</ref>

===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.

===Contours===
Contours are the class of curves on which we define contour integration. A '''contour''' is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves <math>\gamma_1,\dots,\gamma_n</math> be such that the terminal point of <math>\gamma_i</math> coincides with the initial point of <math>\gamma_{i+1}</math> for all <math>i</math> such that <math>1\leq i<n</math> . This includes all directed smooth curves. Also, a single point in the complex plane is considered a contour. The symbol <math>+</math> is often used to denote the piecing of curves together to form a new curve. Thus we could write a contour <math>\Gamma</math> that is made up of <math>n</math> curves as
<math display=block> \Gamma = \gamma_1 + \gamma_2 + \cdots + \gamma_n.</math>