Differential Calculus in the Complex Plane C

In this chapter we shall first give an introduction to complex numbers and their topology. In doing so we shall assume that this is not the first time the reader has encountered the system C of complex numbers. The same assumption is made for topological notions in C (convergence, continuity etc.). For this reason we shall not dwell on these matters. In Sect. I.4 we introduce the notion of complex derivative. One can begin reading directly with this section if one is already sufficiently familiar with the complex numbers and their topology. In Sect. I.5 the relationship between real differentiability and complex differentiability will be treated (the Cauchy-Riemann differential equations).

The story of the complex numbers from their early beginnings in the 16^th century until their eventual full acceptance in the course of the 19^th century—probably by the scientific authority of C.F. GAUSS— as well as the rather lengthy period of uncertainty and unclarity about them, is an impressive example in the history of mathematics. The historically interested reader should read [Re2]. For more historical remarks about complex numbers see also [CE].