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. Complex Variables with Applications......Page 3 Preface......Page 6 Contents......Page 9 1.1 The Complex Field......Page 12 1.2 Rectangular Representation......Page 16 1.3 Polar Representation......Page 26 2.1 Point Sets in the Plane......Page 36 2.2 Sequences......Page 43 2.3 Compactness......Page 50 2.4 Stereographic Projection......Page 55 2.5 Continuity......Page 59 3.1 Basic Mappings......Page 72 3.2 Linear Fractional Transformations......Page 77 3.3 Other Mappings......Page 96 4.1 The Exponential Function......Page 102 4.2 Mapping Properties......Page 111 4.3 The Logarithmic Function......Page 119 4.4 Complex Exponents......Page 125 5.1 Cauchy–Riemann Equation......Page 132 5.2 Analyticity......Page 141 5.3 Harmonic Functions......Page 152 6.1 Sequences Revisited......Page 163 6.2 Uniform Convergence......Page 174 6.3 Maclaurin and Taylor Series......Page 183 6.4 Operations on Power Series......Page 196 7.1 Curves......Page 205 7.2 Parameterizations......Page 217 7.3 Line Integrals......Page 227 7.4 Cauchy’s Theorem......Page 236 8.1 Cauchy’s Integral Formula......Page 253 8.2 Cauchy’s Inequality and Applications......Page 273 8.3 Maximum Modulus Theorem......Page 285 9.1 Laurent Series......Page 295 9.2 Classification of Singularities......Page 303 9.3 Evaluation of Real Integrals......Page 318 9.4 Argument Principle......Page 341 10.1 Comparison with Analytic Functions......Page 358 10.2 Poisson Integral Formula......Page 367 10.3 Positive Harmonic Functions......Page 380 11.1 Conformal Mappings......Page 388 11.2 Normal Families......Page 399 11.3 Riemann Mapping Theorem......Page 404 11.4 The Class S......Page 414 12.1 Infinite Products......Page 420 12.2 Weierstrass’ Product Theorem......Page 431 12.3 Mittag-Leffler Theorem......Page 446 13.1 Basic Concepts......Page 454 13.2 Special Functions......Page 467 References and Further Reading......Page 482 Index of Special Notations......Page 484 Index......Page 487 Hints for Selected Questions and Exercises......Page 493 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