Mapping Contour Integrals
This Demonstration shows the paths of contour integrals over some basic complex functions.[more]
The upper-left panel is the original complex -plane. Use the mouse to draw a contour in the -plane.
The upper-right panel shows the image of the -plane using the complex function selected in the "choose function" popup menu.
The lower-left panel shows the corresponding contour integration of . The contour integrals are not evaluated.[less]
An integral of the complex plane is a line integral over a specified path .
When the path of a complex line integral is a closed curve, the value of the integral can be evaluated using Cauchy's residue theorem, , equal to times the sum of the residues inside the contour.
In this Demonstration, you can see that when the path of integration is closed, its image on the projected plane is also closed. But when the contour contains singularities, the image of the contour is topologically different from the original closed curve.
A singularity changes the number of rotations of the curve about the point; this is called the winding number. When the winding number is , a clockwise loop is transformed to a counterclockwise loop. The contour integration is nonzero only in this case.
Singularities, branch points and branch cuts are shown in pink in the -plane; the singularity with winding number is shown as a magenta point.
 T. Needham, Visual Complex Analysis, Oxford: Clarendon Press, 1998.