This book presents complex analysis in one variable in the context of modern mathematics, with clear connections to several complex variables, de Rham theory, real analysis, and other branches of mathematics. Thus, covering spaces are used explicitly in dealing with Cauchy's theorem, real variable methods are illustrated in the Loman-Menchoff theorem and in the corona theorem, and the algebraic structure of the ring of holomorphic functions is studied.Using the unique position of complex analysis, a field drawing on many disciplines, the book also illustrates powerful mathematical ideas and tools, and requires minimal background material. Cohomological methods are introduced, both in connection with the existence of primitives and in the study of meromorphic functionas on a compact Riemann surface. The proof of Picard's theorem given here illustrates the strong restrictions on holomorphic mappings imposed by curvature conditions.New to this second edition, a collection of over 100 pages worth of exercises, problems, and examples gives students an opportunity to consolidate their command of complex analysis and its relations to other branches of mathematics, including advanced calculus, topology, and real applications.

I Complex Analysis in One Variable.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard's Theorem.- 5 Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem.- 6 Applications of Runge's Theorem.- 7 Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- II Exercises.- 0 Review of Complex Numbers.- 1 Elementary Theory of Holomorphic Functions.- 2 Covering Spaces and the Monodromy Theorem.- 3 The Winding Number and the Residue Theorem.- 4 Picard's Theorem.- 5 The Inhomogeneous Cauchy-Riemann Equation and Runge's Theorem.- 6 Applications of Runge's Theorem.- 7 The Riemann Mapping Theorem and Simple Connectedness in the Plane.- 8 Functions of Several Complex Variables.- 9 Compact Riemann Surfaces.- 10 The Corona Theorem.- 11 Subharmonic Functions and the Dirichlet Problem.- Notes for the exercises.- References for the exercises.