Complex analysis

Abstract

We shall have numerous occasions in the sequel to refer to the theory of functions of a complex variable, and we assume the reader to be familiar with the elements of this theory. We present here an outline of the rudiments of complex analysis, chiefly for convenience of reference, but also to fix notation and terminology. To begin at the beginning, we recall that a domain in the complex plane is a nonempty connected open subset of ℂ, and that a (complex-valued) function f defined on a domain Δ is analytic (or holomorphic) on Δ if its derivative f′ exists at each point of Δ. (More generally, if U is an arbitrary nonempty open subset of ℂ and f is a differentiable complex-valued function defined on U, we shall say that f is locally analytic on U. The study of a locally analytic function f on U reduces at once to the study of the analytic functions obtained by restricting f to the components of U; cf. Proposition 3.9.) We also recall that all of the elementary rules of ordinary real differential calculus hold for analytic functions of a complex variable. Thus the sum, product, and quotient rules for computing derivatives all hold for analytic functions, as does the chain rule. In particular, every complex polynomial function is analytic on the entire complex plane, and may be differentiated by means of the same elementary rule learned in calculus. Similarly, every complex rational function is analytic on the complement of the (finite) set of points at which its denominator vanishes.KeywordsAnalytic ContinuationSimple PolygonParameter IntervalRemovable SingularityTriangular RegionThese 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.