Jerry R. Muir

Complex Analysis

• Gebundenes Buch

Produktbeschreibung

A thorough introduction to the theory of complex functions emphasizing the beauty, power, and counterintuitive nature of the subject

Written with a reader-friendly approach, Complex Analysis: A Modern First Course in Function Theory features a self-contained, concise development of the fundamental principles of complex analysis. After laying groundwork on complex numbers and the calculus and geometric mapping properties of functions of a complex variable, the author uses power series as a unifying theme to define and study the many rich and occasionally surprising properties of analytic functions, including the Cauchy theory and residue theorem. The book concludes with a treatment of harmonic functions and an epilogue on the Riemann mapping theorem.

Thoroughly classroom tested at multiple universities, Complex Analysis: A Modern First Course in Function Theory features:

Plentiful exercises, both computational and theoretical, of varying levels of difficulty, including several that could be used for student projects
Numerous figures to illustrate geometric concepts and constructions used in proofs
Remarks at the conclusion of each section that place the main concepts in context, compare and contrast results with the calculus of real functions, and provide historical notes
Appendices on the basics of sets and functions and a handful of useful results from advanced calculus Appropriate for students majoring in pure or applied mathematics as well as physics or engineering, Complex Analysis: A Modern First Course in Function Theory is an ideal textbook for a one-semester course in complex analysis for those with a strong foundation in multivariable calculus. The logically complete book also serves as a key reference for mathematicians, physicists, and engineers and is an excellent source for anyone interested in independently learning or reviewing the beautiful subject of complex analysis.

Produktdetails

• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 276
• Erscheinungstermin: 24. April 2015
• Englisch
• Abmessung: 240mm x 161mm x 19mm
• Gewicht: 583g
• ISBN-13: 9781118705223
• ISBN-10: 111870522X
• Artikelnr.: 41023746

Autorenporträt

Jerry R. Muir, Jr., PhD, is Professor of Mathematics at The University of Scranton. He has authored over one dozen research articles in complex-flavored analysis, primarily on geometric function theory in several complex variables.

Inhaltsangabe

Preface ix 1 The Complex Numbers 1 1.1 Why? 1 1.2 The Algebra of Complex
Numbers 3 1.3 The Geometry of the Complex Plane 7 1.4 The Topology of the
Complex Plane 9 1.5 The Extended Complex Plane 16 1.6 Complex Sequences 18
1.7 Complex Series 24 2 Complex Functions and Mappings 29 2.1 Continuous
Functions 29 2.2 Uniform Convergence 34 2.3 Power Series 38 2.4 Elementary
Functions and Euler's Formula 43 2.5 Continuous Functions as Mappings 50
2.6 Linear Fractional Transformations 53 2.7 Derivatives 64 2.8 The
Calculus of Real Variable Functions 70 2.9 Contour Integrals 75 3 Analytic
Functions 87 3.1 The Principle of Analyticity 87 3.2 Differentiable
Functions are Analytic 89 3.3 Consequences of Goursat's Theorem 100 3.4 The
Zeros of Analytic Functions 104 3.5 The Open Mapping Theorem and Maximum
Principle 108 3.6 The Cauchy-Riemann Equations 113 3.7 Conformal Mapping
and Local Univalence 117 4 Cauchy's Integral Theory 127 4.1 The Index of a
Closed Contour 127 4.2 The Cauchy Integral Formula 133 4.3 Cauchy's Theorem
139 5 The Residue Theorem 145 5.1 Laurent Series 145 5.2 Classification of
Singularities 152 5.3 Residues 158 5.4 Evaluation of Real Integrals 165 5.5
The Laplace Transform 174 6 Harmonic Functions and Fourier Series 183 6.1
Harmonic Functions 183 6.2 The Poisson Integral Formula 191 6.3 Further
Connections to Analytic Functions 201 6.4 Fourier Series 210 Epilogue 227 A
Sets and Functions 239 B Topics from Advanced Calculus 247 References 255
Index 257