Complex Variables with Applications

Complex numbers can be viewed in several ways: as an element in a field, as a point in the plane, and as a two-dimensional vector. Examined properly, each perspective provides crucial insight into the interrelations between the complex number system and its parent, the real number system. The authors explore these relationships by adopting both generalization and specialization methods to move from real variables to complex variables, and vice versa, while simultaneously examining their analytic and geometric characteristics, using geometry to illustrate analytic concepts and employing analysis to unravel geometric notions. The engaging exposition is replete with discussions, remarks, questions, and exercises, motivating not only understanding on the part of the reader, but also developing the tools needed to think critically about mathematical problems. This focus involves a careful examination of the methods and assumptions underlying various alternative routes that lead to the same destination. The material includes numerous examples and applications relevant to engineering students, along with some techniques to evaluate various types of integrals. The book may serve as a text for an undergraduate course in complex variables designed for scientists and engineers or for mathematics majors interested in further pursuing the general theory of complex analysis. The only prerequistite is a basic knowledge of advanced calculus. The presentation is also ideally suited for self-study. front-matter......Page 2 1Algebraic and Geometric Preliminaries......Page 12 2Topological and Analytic Preliminaries......Page 36 3Bilinear Transformations and Mappings......Page 72 4Elementary Functions......Page 102 5Analytic Functions......Page 132 6Power Series......Page 163 7Complex Integration and Cauchy’s Theorem......Page 205 8Applications of Cauchy’s Theorem......Page 253 9Laurent Series and the Residue Theorem......Page 295 back-matter......Page 358 10Harmonic Functions......Page 398 11Conformal Mapping and the Riemann Mapping Theorem......Page 428 12Entire and Meromorphic Functions......Page 460 13Analytic Continuation......Page 494 Explores the interrelations between real and complex numbers by adopting both generalization and specialization methods to move between them, while simultaneously examining their analytic and geometric characteristics Engaging exposition with discussions, remarks, questions, and exercises to motivate understanding and critical thinking skills Encludes numerous examples and applications relevant to science and engineering students "Complex Analysis with Applications in Science and Engineering weaves together theory and extensive applications in mathematics, physics and engineering. In this edition there are many new problems, revised sections, and an entirely new chapter on analytic continuation. This work will serve as a textbook for undergraduate and graduate students in the areas noted above."--Publisher's website