Local Properties of holomorphic functions

Abstract

AbstractEvery student remembers from his first (or, at least, second) course on calculus that the function $$f\,{\text{:}} \mathbb{R} \to \mathbb{R}{\text{, }}x \mapsto \left\{ {\begin{array}{*{20}c} {\quad 0,} & {{\text{if }}x = 0} \\ {e^{ - \frac{1} {{x^2 }}} ,} & {{\text{if }}x \ne 0} \\ \end{array} } \right.$$ is of class C∞, but its Taylor series represents f only at the origin, since f(k) (0) = 0 for all k ∈ ℕ. One major difference between real and complex analysis is that a holomorphic function on a domain is determined completely by local information. If two holomorphic functions f, \(g \in \mathcal{O}\) (D) coincide on an open subset U of a domain D they coincide on all of D by the Identity Theorem. Locally, a holomorphic function is represented by its Taylor series. In this final chapter we study holomorphic functions from this local point of view, i.e., we do not take into account the domain of definition of a function, but only its local representation. This leads to the concept of germ of a holomorphic function. An equivalent approach is to examine the rings ℂ {z1, . . . , zn} resp. ℂ [[z1, . . . , zn]] of convergent resp. of formal power series and to study algebraic properties of these rings. The reader not familiar with the algebraic concepts needed in this chapter may refer to [7].