by M W Wong (York University, Canada)

Table of Contents (62k)
Preface (51k)
Chapter 1: Complex Numbers (152k)

This book is ideal for a one-semester course for advanced undergraduate students and first-year graduate students in mathematics. It is a straightforward and coherent account of a body of knowledge in complex analysis, from complex numbers to Cauchy's integral theorems and formulas to more advanced topics such as automorphism groups, the Schwarz problem in partial differential equations, and boundary behavior of harmonic functions.

The book covers a wide range of topics, from the most basic complex numbers to those that underpin current research on some aspects of analysis and partial differential equations. The novelty of this book lies in its choice of topics, genesis of presentation, and lucidity of exposition.

• Complex Numbers
• Arguments and Polar Forms of Complex Numbers
• Exponentials, Powers and Roots
• Functions of a Complex Variable
• Holomorphic Functions and Cauchy–Riemann Equations
• The Exponential, Trigonometric and Hyperbolic Functions
• Logarithms, Complex Powers, Branches and Cuts
• Contour Integrals and Path Independence
• Cauchy's Integral Theorems
• Cauchy's Integral Formulas
• Taylor Series and Power Series
• Laurent Series and Isolated Singularities
• Residues
• Trigonometric Integrals: Cauchy Principal Values of Improper Integrals on (–∞, ∞)
• Fourier Transforms of Rational Functions
• Singular Integrals on (–∞, ∞)
• Integrals on Branch Cuts
• Biholomorphisms
• Zeros, Maximum Modulus Principle and Schwarz's Lemma
• Aut(Δ) and SU(1,1)
• Aut(H), SL(2, P) and the Iwasawa Decomposition
• Harmonic Functions and the Schwarz Problem on Δ