Complex Variables A Physical Approach with Applications

The idea of complex numbers dates back at least 300 years—to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers.This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications. Features:This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysisThis book has an exceptionally large number of examples and a large number of figures. The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking.Incisive applications appear throughout the book.Partial differential equations are used as a unifying theme.

Basic IdeasThe Exponential and ApplicationsHolomorphic and Harmonic FunctionsThe Cauchy TheoryApplications of the Cauchy TheoryIsolated SingularitiesMeromorphic FunctionsThe Calculus of ResiduesThe Argument PrincipleThe Maximum PrincipleThe Geometric TheoryApplications of Conformal Mapping Harmonic FunctionsThe Fourier TheoryOther TransformsBoundary Value Problems

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 130 books and more than 250 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.