Complex Analysis / Edition 1

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Overview

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.

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Product Details

  • ISBN-13: 9780387950693
  • Publisher: Springer New York
  • Publication date: 5/18/2001
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: 1st ed. 2001. Corr. 2nd printing 2003
  • Edition number: 1
  • Pages: 478
  • Product dimensions: 1.00 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface
Introduction
Ch. I The Complex Plane and Elementary Functions 1
Ch. II Analytic Functions 33
Ch. III Line Integrals and Harmonic Functions 70
Ch. IV Complex Integration and Analyticity 102
Ch. V Power Series 130
Ch. VI Laurent Series and Isolated Singularities 165
Ch. VII The Residue Calculus 195
Ch. VIII The Logarithmic Integral 224
Ch. IX The Schwarz Lemma and Hyperbolic Geometry 260
Ch. X Harmonic Functions and the Reflection Principle 274
Ch. XI Conformal Mapping 289
Ch. XII Compact Families of Meromorphic Functions 315
Ch. XIII Approximation Theorems 342
Ch. XIV Some Special Functions 361
Ch. XV The Dirichlet Problem 390
Ch. XVI Riemann Surfaces 418
Hints and Solutions for Selected Exercises 447
References 469
List of Symbols 471
Index 473
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