# Complex Variables with Applications

Springer Science & Business Media, May 26, 2007 - Mathematics - 514 pages

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.

### What people are saying -Write a review

User Review - Flag as inappropriate

A wonderful book.

### Contents

 Algebraic and Geometric Preliminaries 1 12 Rectangular Representation 5 13 Polar Representation 15 Topological and Analytic Preliminaries 25 22 Sequences 32 23 Compactness 39 24 Stereographic Projection 44 25 Continuity 48
 74 Cauchys Theorem 226 Applications of Cauchys Theorem 243 82 Cauchys Inequality and Applications 263 83 Maximum Modulus Theorem 275 Laurent Series and the Residue Theorem 285 92 Classification of Singularities 293 93 Evaluation of Real Integrals 308 94 Argument Principle 331

 Bilinear Transformations and Mappings 61 32 Linear Fractional Transformations 66 33 Other Mappings 85 Elementary Functions 91 42 Mapping Properties 100 43 The Logarithmic Function 108 44 Complex Exponents 114 Analytic Functions 121 52 Analyticity 130 53 Harmonic Functions 141 Power Series 153 62 Uniform Convergence 164 63 Maclaurin and Taylor Series 173 64 Operations on Power Series 186 Complex Integration and Cauchys Theorem 195 72 Parameterizations 207 73 Line Integrals 217
 Harmonic Functions 348 102 Poisson Integral Formula 358 103 Positive Harmonic Functions 371 Conformal Mapping and the Riemann Mapping Theorem 379 112 Normal Families 390 113 Riemann Mapping Theorem 395 114 The Class S 405 Entire and Meromorphic Functions 411 122 Weierstrass Product Theorem 422 123 MittagLeffler Theorem 437 Analytic Continuation 445 132 Special Functions 458 References and Further Reading 473 Index of Special Notations 475 Index 479 Hints for Selected Questions and Exercises 485 Copyright