Complex Analysis

This chapter extends the homotopy versions of Cauchy's Theorem in Chapter 9 to consider how 'holes' in the domain caused by singularities affect complex integrals that wind round them. It uses a 'bare hands' approach based on step paths guaranteed by the Paving Lemma. A sum of closed rectangles in the domain is a cycle; if the interior of each rectangle also lies in the domain, the cycle is called a boundary. The homology version of Cauchy's Theorem shows that the integral round a boundary is zero.