This authoritative text presents the classical theory of functions of a single complex variable in complete mathematical and historical detail. Requiring only minimal, undergraduate-level prerequisites, it covers the fundamental areas of the subject with depth, precision, and rigor. Standard and novel proofs are explored in unusual detail, and exercises - many with helpful hints - provide ample opportunities for practice and a deeper understanding of the material.

In addition to the mathematical theory, the author also explores how key ideas in complex analysis have evolved over many centuries, allowing readers to acquire an extensive view of the subject's development. Historical notes are incorporated throughout, and a bibliography containing more than 2,000 entries provides an exhaustive list of both important and overlooked works.

Classical Analysis in the Complex Plane will be a definitive reference for both graduate studentsand experienced mathematicians alike, as well as an exemplary resource for anyone doing scholarly work in complex analysis. The author's expansive knowledge of and passion for the material is evident on every page, as is his desire to impart a lasting appreciation for the subject.

"I can honestly say that Robert Burckel's book has profoundly influenced my view of the subject of complex analysis. It has given me a sense of the historical flow of ideas, and has acquainted me with byways and ancillary results that I never would have encountered in the ordinary course of my work. The care exercised in each of his proofs is a model of clarity in mathematical writing...Anyone in the field should have this book on [their bookshelves] as a resource and an inspiration."

• From the Foreword by Steven G. Krantz

Autorentext

Robert B. Burckel is Professor Emeritus of Mathematics at Kansas State University.

Inhalt

Prerequisites and Preliminaries.- Curves, Connectedness and Convexity.- (Complex) Derivative and (Curvilinear) Integrals.- Power Series and the Exponential Function.- The Index and some Plane Topology.-Consequences of the Cauchy-Goursat Theorem-Maximum Principles and the Local Theory.- Schwarz' Lemma and its Many Applications.- Convergent Sequences of Holomorphic Functions.- Polynomial and Rational Approximation-Runge Theory.- The Riemann Mapping Theorem.- Simple and Double Connectivity.- Isolated Singularities.- Omitted Values and Normal Families.- Bibliography.- Name Index.- Subject Index.- Symbol Index.- Index of Series Summed and Integrals Evaluated.