Introduction to Complex Analysis (Graduate Studies in Mathematics)

Main subject categories: • Complex analysis • Complex numbers • Fourier analysis • Complex function theory • Residue calculus • Conformal maps • Elliptic functions • Elliptic integrals • Differential equationsIn this text, the reader will learn that all the basic functions that arise in calculus such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions.Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here.This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus. In this text, the reader will learn that all the basic functions that arise in calculus such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here. This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus. Cover Contents Preface Some basic notation Chapter 1. Basic calculus in the complex domain Chapter 2. Going deeper – the Cauchy integral theorem and consequences Chapter 3. Fourier analysis and complex function theory Chapter 4. Residue calculus, the argument principle, and two very special functions Chapter 5. Conformal maps and geometrical aspects of complex function theory Chapter 6. Elliptic functions and elliptic integrals Chapter 7. Complex analysis and differential equations Appendix A. Complementary material Bibliography Index Michael E. Taylor. Includes Bibliographical References And Index.