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Handbook of Complex Variables - Steven G. Krantz

Handbook of Complex Variables

Hardcover Published: 14th October 1999
ISBN: 9780817640118
Number Of Pages: 290

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This book is a convenient reference for the scientist, student, or engineer whose work necessitates the use of the basic concepts in complex analysis. Rather than being a book of mathematical theory; it is instead a book of practice in which all the ideas and applications of complex analysis and differential equations are treated, as well as the applicable computational software.

Industry Reviews

"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

Preface
List of Figures
The Complex Planep. 1
Complex Arithmeticp. 1
The Exponential and Applicationsp. 7
Holomorphic Functionsp. 12
The Relationship of Holomorphic and Harmonic Functionsp. 16
Complex Line Integralsp. 19
Real and Complex Line Integralsp. 19
Complex Differentiability and Conformalityp. 23
The Cauchy Integral Thoerem and Formulap. 26
A Coda on the Limitations of the Cauchy Integral Formulap. 28
Applications of the Cauchy Theoryp. 31
The Derivatives of a Holomorphic Functionp. 31
The Zeros of Holomorphic Functionp. 36
Isolated Singularities and Laurent Seriesp. 41
The Behavior of a Holomorphic Function near an Isolated Singularityp. 41
Expansion around Singular Pointsp. 43
Examples of Laurent Expansionsp. 46
The Calculus of Residuesp. 48
Applications to the Calculation of Definite Integrals and Sumsp. 51
Meromorphic Functions and Singularities at Infinityp. 63
The Argument Principlep. 69
Counting Zeros and Polesp. 69
The Local Geometry of Holomorphic Functionsp. 73
Further Results on the Zeros of Holomorphic Functionsp. 74
The Maximum Principlep. 76
The Schwarz Lemmap. 77
The Geometric Theory of Holomorphic Functionsp. 79
The Idea of a Conformal Mappingp. 79
Conformal Mappings of the Unit Discp. 80
Linear Fractional Transformationsp. 81
The Riemann Mapping Theoremp. 86
Conformal Mappings of Annulip. 87
Harmonic Functionsp. 89
Basic Properties of Harmonic Functionsp. 89
The Maximum Principle and the Mean Value Propertyp. 91
The Poisson Integral Formulap. 92
Regularity of Harmonic Functionsp. 94
The Schwarz Reflection Principlep. 95
Harnack's Principlep. 97
The Dirichlet Problem and Subharmonic Functionsp. 97
The General Solution of the Dirichlet Problemp. 101
Infinite Series and Productsp. 103
Basic Concepts Concerning Infinite Sums and Productsp. 103
The Weierstrass Factorization Theoremp. 109
The Theorems of Weierstrass and Mittag-Lefflerp. 110
Normal Familiesp. 113
Applications of Infinite Sums and Productsp. 117
Jensen's Formula and an Introduction to Blaschke Productsp. 117
The Hadamard Gap Theoremp. 119
Entire Functions of Finite Orderp. 120
Analytic Continuationp. 123
Definition of an Analytic Function Elementp. 123
Analytic Continuation along a Curvep. 130
The Monodromy Theoremp. 131
The Idea of Riemann Surfacep. 135
Picard's Theoremsp. 140
Rational Approximation Theoryp. 143
Runge's Theoremp. 143
Mergelyan's Theoremp. 146
Special Classes of Holomorphic Functionsp. 149
Schlicht Functions and the Bieberbach Conjecturep. 149
Extension to the Boundary of Conformal Mappingsp. 151
Hardy Spacesp. 152
Special Functionsp. 155
Introductionp. 155
The Gamma and Beta Functionsp. 155
Riemann's Zeta Functionp. 158
Some Counting Functions and a Few Technical Lemmasp. 162
Applications that Depend on Conformal Mappingp. 163
Conformal Mappingp. 163
Application of Conformal Mapping to the Dirichlet Problemp. 164
Physical Examples Solved by Means of Conformal Mappingp. 168
Numerical Techniques of Conformal Mappingp. 175
Transform Theoryp. 195
Introductory Remarksp. 195
Fourier Seriesp. 195
The Fourier Transformp. 202
The Laplace Transformp. 212
The z-Transformp. 214
Computer Packages for Studying Complex Variablesp. 219
Introductory Remarksp. 219
The Software Packagesp. 219
Glossary of Terms from Complex Variable Theory and Analysisp. 231
List of Notationp. 269
Table of Laplace Transformsp. 273
A Guide to the Literaturep. 275
Referencesp. 279
Indexp. 283
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780817640118
ISBN-10: 0817640118
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 290
Published: 14th October 1999
Publisher: BIRKHAUSER BOSTON INC
Country of Publication: US
Dimensions (cm): 22.86 x 15.24  x 2.54
Weight (kg): 0.57

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