Complex Variables with Applications
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Description: 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.
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All the information you need in one place! Each Study Brief is a summary of one specific subject; facts, figures, and explanations to help you learn faster.
List price: $69.95
Copyright year: 2006
Publisher: Birkhauser Boston
Publication date: 9/13/2006
Size: 6.00" wide x 9.25" long x 1.00" tall
|Algebraic and Geometric Preliminaries|
|The Complex Field|
|Topological and Analytic Preliminaries|
|Point Sets in the Plane|
|Bilinear Transformations and Mappings|
|Linear Fractional Transformations|
|The Exponential Function|
|The Logarithmic Function|
|Maclaurin and Taylor Series|
|Operations on Power Series|
|Complex Integration and Cauchy's Theorem|
|Applications of Cauchy's Theorem|
|Cauchy's Integral Formula|
|Cauchy's Inequality and Applications|
|Maximum Modulus Theorem|
|Laurent Series and the Residue Theorem|
|Classification of Singularities|
|Evaluation of Real Integrals|
|Comparison with Analytic Functions|
|Poisson Integral Formula|
|Positive Harmonic Functions|
|Conformal Mapping and the Riemann Mapping Theorem|
|Riemann Mapping Theorem|
|The Class S|
|Entire and Meromorphic Functions|
|Weierstrass' Product Theorem|
|References and Further Reading|
|Index of Special Notations|
|Hints for Selected Questions and Exercises|