Complex Variables with Applications

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

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About the author (2007)

Herb Silverman is Founder and President of the Secular CoaliHerb Silverman is Founder and President of the Secular Coalition for America, and Distinguished Professor Emeritus of Mation for America, and Distinguished Professor Emeritus of Mathematics at the College of Charleston. He ran for governor thematics at the College of Charleston. He ran for governor of South Carolina in the 1990s to challenge a state law thatof South Carolina in the 1990s to challenge a state law that required religious belief to hold public office. After an e required religious belief to hold public office. After an eight-year battle, Herb won a unanimous decision in the Southight-year battle, Herb won a unanimous decision in the South Carolina Supreme Court, which struck down this religious te Carolina Supreme Court, which struck down this religious test requirement. st requirement.

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