Complex Analysis

An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.

Table of Contents

FIRST PART

Chapter I The Complex Plane and Elementary Functions

Chapter II Analytic Functions

Chapter III Line Integrals and Harmonic Functions

Chapter IV Complex Integration and Analyticity

Chapter V Power Series

Chapter VI Laurent Series and Isolated Singularities

Chapter VII The Residue Calculus

SECOND PART

Chapter VIII The Logarithmic Integral

Chapter IX The Schwarz Lemma and Hyperbolic Geometry

Chapter X Harmonic Functions and the Reflection Principle

Chapter XI Conformal Mapping

THIRD PART

Chapter XII Compact Families of Meromorphic Functions

Chapter XIII Approximation Theorems

Chapter XIV Some Special Functions

Chapter XV The Dirichlet Problem

Chapter XVI Riemann Surfaces