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:
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:
Noah Dana-Picard 2007-12-24