Let be a path of integration. It is smooth at a point if is derivable at and if its first derivative is continuous at . The path is smooth if it is smooth at every point; it is smooth by parts if it is not smooth at only a finite number of points.

A **loop** is an integrating path whose origin and endpoints are
identical. If it is smooth and does not intersect itself at another
point, we will call it a **Jordan curve.** If we travel exactly once along the loop, we will call it a simple loop.
The integral of a function
along a simple loop
will be denoted as follows:

Note the little circle on the integration symbol.

A Jordan curve determines in the plane three disjoint regions: the curve itself, the interior (= a bounded region) and the exterior (= an unbounded region).

Note that the converse is not true: if is not analytic on the interior of , the integral can either vanish or not. For example, compute the following integral:

Using the parametrization , we can show that . The integral vanishes, despite the fact that the function fails to be analytic at 0, which is an interior point of the unit circle (defined here by the equation ).

- , for every Jordan curve in the Cauchy-Argand plane.
- Let and let be the unit circle. As is analytic on the closed unit disk, we have: .

Noah Dana-Picard 2007-12-24