Handbook of Complex Variables / Edition 1

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Handbook of Complex Variables is a reference work for scientists and engineers who need to know and use essential information and methods involving complex variables and analysis. Its focus is on basic concepts and informational tools for mathematical "practice": solving problems in applied mathematics, science, and engineering." "This handbook is a reference and authoritative resource for all professionals, practitioners, and researchers in mathematics, physical science, and engineering.
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Editorial Reviews

From the Publisher
"This modern book can be warmly recommended to mathematicians as well as to users of applied texts in complex analysis; in particular it will be useful to students preparing for an examination in the subject." —Mathematical Reviews

"Creating a 'handbook' such as this is an interesting concept, and to this reviewer’s knowledge this is the only one of its type in complex analysis. . . . This book may well be timely and useful to the readers it is intended for: working scientists, students, and engineers . . . The topics contained are quite broad . . . It is noteworthy that a glossary is included that provides the reader with a useful guide to terminology and basic concepts. Other valuable features are: (1) a discussion of the available computer packages that can do some complex analysis such as Maple and Mathematica, (2) a pictorial catalog of conformal well-known maps, and (3) tables of Laplace transforms." —SIAM Review

"Krantz...has two audiences in mind for this handbook: first, the working scientist, with no background in complex analysis, who seeks a specific result to solve a specific problem; and second, the mathematician or scientist who once studied complex analysis and now seeks a compendium of results as an aid to memory. Though Krantz warns that this handbook contains no theory...and thus cannot serve as a textbook, the undergraduate student of complex analysis will nevertheless find certain sections replete with instructive examples (e.g., applications of contour integrations to definite integrals and sums; conformal mapping). Also, the glossary of terminology and notation should offer a useful aid to study.... Students should also see the chapter devoted to surveying computer packages for the study of complex variables. In an undergraduate library, this book can be counted as a supplement to an otherwise strong collection in functions of a single complex variable." —Choice

"This handbook of complex variables is a comprehensive references work for scientists, students and engineers who need to know and use the basic concepts in complex analysis of one variable. It is not a book of mathematical theory but a book of mathematical practice. All basic ideas of complex analysis and many typical applications are treated. It is also written in a very vivid style and it contains many helpful figures and graphs." —-Zentralblatt MATH

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Product Details

  • ISBN-13: 9780817640118
  • Publisher: Birkhauser Verlag
  • Publication date: 10/1/1999
  • Edition description: 1999
  • Edition number: 1
  • Pages: 290
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

List of Figures
1 The Complex Plane 1
1.1 Complex Arithmetic 1
1.2 The Exponential and Applications 7
1.3 Holomorphic Functions 12
1.4 The Relationship of Holomorphic and Harmonic Functions 16
2 Complex Line Integrals 19
2.1 Real and Complex Line Integrals 19
2.2 Complex Differentiability and Conformality 23
2.3 The Cauchy Integral Thoerem and Formula 26
2.4 A Coda on the Limitations of the Cauchy Integral Formula 28
3 Applications of the Cauchy Theory 31
3.1 The Derivatives of a Holomorphic Function 31
3.2 The Zeros of Holomorphic Function 36
4 Isolated Singularities and Laurent Series 41
4.1 The Behavior of a Holomorphic Function near an Isolated Singularity 41
4.2 Expansion around Singular Points 43
4.3 Examples of Laurent Expansions 46
4.4 The Calculus of Residues 48
4.5 Applications to the Calculation of Definite Integrals and Sums 51
4.6 Meromorphic Functions and Singularities at Infinity 63
5 The Argument Principle 69
5.1 Counting Zeros and Poles 69
5.2 The Local Geometry of Holomorphic Functions 73
5.3 Further Results on the Zeros of Holomorphic Functions 74
5.4 The Maximum Principle 76
5.5 The Schwarz Lemma 77
6 The Geometric Theory of Holomorphic Functions 79
6.1 The Idea of a Conformal Mapping 79
6.2 Conformal Mappings of the Unit Disc 80
6.3 Linear Fractional Transformations 81
6.4 The Riemann Mapping Theorem 86
6.5 Conformal Mappings of Annuli 87
7 Harmonic Functions 89
7.1 Basic Properties of Harmonic Functions 89
7.2 The Maximum Principle and the Mean Value Property 91
7.3 The Poisson Integral Formula 92
7.4 Regularity of Harmonic Functions 94
7.5 The Schwarz Reflection Principle 95
7.6 Harnack's Principle 97
7.7 The Dirichlet Problem and Subharmonic Functions 97
7.8 The General Solution of the Dirichlet Problem 101
8 Infinite Series and Products 103
8.1 Basic Concepts Concerning Infinite Sums and Products 103
8.2 The Weierstrass Factorization Theorem 109
8.3 The Theorems of Weierstrass and Mittag-Leffler 110
8.4 Normal Families 113
9 Applications of Infinite Sums and Products 117
9.1 Jensen's Formula and an Introduction to Blaschke Products 117
9.2 The Hadamard Gap Theorem 119
9.3 Entire Functions of Finite Order 120
10 Analytic Continuation 123
10.1 Definition of an Analytic Function Element 123
10.2 Analytic Continuation along a Curve 130
10.3 The Monodromy Theorem 131
10.4 The Idea of Riemann Surface 135
10.5 Picard's Theorems 140
11 Rational Approximation Theory 143
11.1 Runge's Theorem 143
11.2 Mergelyan's Theorem 146
12 Special Classes of Holomorphic Functions 149
12.1 Schlicht Functions and the Bieberbach Conjecture 149
12.2 Extension to the Boundary of Conformal Mappings 151
12.3 Hardy Spaces 152
13 Special Functions 155
13.0 Introduction 155
13.1 The Gamma and Beta Functions 155
13.2 Riemann's Zeta Function 158
13.3 Some Counting Functions and a Few Technical Lemmas 162
14 Applications that Depend on Conformal Mapping 163
14.1 Conformal Mapping 163
14.2 Application of Conformal Mapping to the Dirichlet Problem 164
14.3 Physical Examples Solved by Means of Conformal Mapping 168
14.4 Numerical Techniques of Conformal Mapping 175
15 Transform Theory 195
15.0 Introductory Remarks 195
15.1 Fourier Series 195
15.2 The Fourier Transform 202
15.3 The Laplace Transform 212
15.4 The z-Transform 214
16 Computer Packages for Studying Complex Variables 219
16.0 Introductory Remarks 219
16.1 The Software Packages 219
Glossary of Terms from Complex Variable Theory and Analysis 231
List of Notation 269
Table of Laplace Transforms 273
A Guide to the Literature 275
References 279
Index 283
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