The Geometry of Integrating a Power around the Origin

This Demonstration shows some geometric relationships between terms in the contour integral around the origin of , . Note in particular that the complete integral around the origin takes on a nonzero value only when . In this case the term (green) and the term (red) rotate in such in such a way that their product (green) points in a fixed direction. In all other cases, the product rotates an integer number of times along the complete contour, resulting in a zero value.

The term in the legend refers to the end-point of the (black) arc of integration.

Computing the integral of around the origin longhand provides a nice complement to this Demonstration. See details on pages 240–241 of the excellent book Complex Analysis with Mathematica by William T. Shaw.