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. Cover page......Page 1 Title page......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 164 6.2 Uniform Convergence......Page 175 6.3 Maclaurin and Taylor Series......Page 184 6.4 Operations on Power Series......Page 197 7.1 Curves......Page 206 7.2 Parameterizations......Page 218 7.3 Line Integrals......Page 228 7.4 Cauchy’s Theorem......Page 237 8.1 Cauchy’s Integral Formula......Page 254 8.2 Cauchy’s Inequality and Applications......Page 274 8.3 Maximum Modulus Theorem......Page 286 9.1 Laurent Series......Page 296 9.2 Classification of Singularities......Page 304 9.3 Evaluation of Real Integrals......Page 319 9.4 Argument Principle......Page 342 10.1 Comparison with Analytic Functions......Page 360 10.2 Poisson Integral Formula......Page 369 10.3 Positive Harmonic Functions......Page 382 11.1 Conformal Mappings......Page 390 11.2 Normal Families......Page 401 11.3 Riemann Mapping Theorem......Page 406 11.4 The Class S......Page 416 12.1 Infinite Products......Page 422 12.2 Weierstrass’ Product Theorem......Page 433 12.3 Mittag-Leffler Theorem......Page 448 13.1 Basic Concepts......Page 456 13.2 Special Functions......Page 469 References and Further Reading......Page 484 Index of Special Notations......Page 486 Index......Page 490 Hints for Selected Questions and Exercises......Page 496 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