Complex Variables with Applications

ISBN-10: 0817644571
ISBN-13: 9780817644574
Edition: 2006
List price: $69.95 Buy it from $19.65
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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  More...

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Book details

List price: $69.95
Copyright year: 2006
Publisher: Birkhauser Boston
Publication date: 9/13/2006
Binding: Hardcover
Pages: 514
Size: 6.00" wide x 9.25" long x 1.00" tall
Weight: 1.892
Language: English

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.

Preface
Algebraic and Geometric Preliminaries
The Complex Field
Rectangular Representation
Polar Representation
Topological and Analytic Preliminaries
Point Sets in the Plane
Sequences
Compactness
Stereographic Projection
Continuity
Bilinear Transformations and Mappings
Basic Mappings
Linear Fractional Transformations
Other Mappings
Elementary Functions
The Exponential Function
Mapping Properties
The Logarithmic Function
Complex Exponents
Analytic Functions
Cauchy-Riemann Equation
Analyticity
Harmonic Functions
Power Series
Sequences Revisited
Uniform Convergence
Maclaurin and Taylor Series
Operations on Power Series
Complex Integration and Cauchy's Theorem
Curves
Parameterizations
Line Integrals
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
Laurent Series
Classification of Singularities
Evaluation of Real Integrals
Argument Principle
Harmonic Functions
Comparison with Analytic Functions
Poisson Integral Formula
Positive Harmonic Functions
Conformal Mapping and the Riemann Mapping Theorem
Conformal Mappings
Normal Families
Riemann Mapping Theorem
The Class S
Entire and Meromorphic Functions
Infinite Products
Weierstrass' Product Theorem
Mittag-Leffler Theorem
Analytic Continuation
Basic Concepts
Special Functions
References and Further Reading
Index of Special Notations
Index
Hints for Selected Questions and Exercises

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