Analysis in the complex plane

The fundamental concept of holomorphic function is introduced via complex differentiability in section I.1. The relation between real and complex differentiability is then discussed, leading to the characterization of holomorphic functions by the Cauchy-Riemann differential equations (I.2). Power series are important examples of holomorphic functions (I.3); we here apply real analysis to show their holomorphy, although Chapter II will open a simpler way. In particular, the real exponential and trigonometric functions can be extended via power series to holomorphic functions on the whole complex plane; we discuss these functions without recurring to the corresponding real theory (I.4). Section I.5 presents an essential tool of complex analysis, viz. integration along paths in the plane. In I.6 we carry over the basic theory to functions of several complex variables. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.