Complex Analysis / Edition 1

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The book provides an introduction to complex analysis for students wit h some familiarity with complex numbers from high school. The book con sists of three parts. The first part comprises the basic core of a cou rse in complex analysis for junior and senior undergraduates. The seco nd part includes various more specialized topics as the argument princ iple, the Schwarz lemma and hyperbolic geometry, the Poisson integral, and the Riemann mapping theorem. The third part consists of a selecti on of topics designed to complete the coverage of all background neces sary for passing Ph.D. qualifying exams in complex analysis. Topics se lected include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces. The three geometries, spherical, euclidean, and hyperbolic, are stressed. Exercises range from the very simple to the quite challe nging, in all chapters. The book is based on lectures given over the y ears by the author at several places.

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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
  • Sales rank: 738,881
  • Product dimensions: 1.00 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

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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