Complex Variables / Edition 2

Complex Variables / Edition 2

2.3 3
by Stephen D. Fisher
     
 

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ISBN-10: 0486406792

ISBN-13: 9780486406794

Pub. Date: 02/16/1999

Publisher: Dover Publications

The most important topics in the theory and application of complex variables receive a thorough, coherent treatment in this introductory text. Intended for undergraduates or graduate students in science, mathematics, and engineering, this volume features hundreds of solved examples, exercises, and applications designed to foster a complete understanding of complex

Overview

The most important topics in the theory and application of complex variables receive a thorough, coherent treatment in this introductory text. Intended for undergraduates or graduate students in science, mathematics, and engineering, this volume features hundreds of solved examples, exercises, and applications designed to foster a complete understanding of complex variables as well as an appreciation of their mathematical beauty and elegance.
Prerequisites are minimal; a three-semester course in calculus will suffice to prepare students for discussions of these topics: the complex plane, basic properties of analytic functions (including a rewritten and reorganized discussion of Cauchy's Theorem), analytic functions as mappings, analytic and harmonic functions in applications, and transform methods. Useful appendixes include tables of conformal mappings and Laplace transforms, as well as solutions to odd-numbered exercises.
Students and teachers alike will find this volume, with its well-organized text and clear, concise proofs, an outstanding introduction to the intricacies of complex variables.

Product Details

ISBN-13:
9780486406794
Publisher:
Dover Publications
Publication date:
02/16/1999
Series:
Dover Books on Mathematics Series
Edition description:
REV
Pages:
448
Sales rank:
476,584
Product dimensions:
6.50(w) x 9.20(h) x 1.00(d)

Table of Contents

1. The complex plane
  1.1 The complex numbers and the complex plane
    1.1.1 A formal view of the complex numbers
  1.2 Some geometry
  1.3 Subsets of the plane
  1.4 Functions and limits
  1.5 The exponential, logarithm, and trigonometric functions
  1.6 Line integrals and Green's theorem
2. Basic properties of analytic functions
  2.1 Analytic and harmonic functions; the Cauchy-Riemann equations
    2.1.1 Flows, fields, and analytic functions
  2.2 Power series
  2.3 Cauchy's theorem and Cauchy's formula
    2.3.1 The Cauchy-Goursat theorem
  2.4 Consequences of Cauchy's formula
  2.5 Isolated singularities
  2.6 The residue theorem and its application to the evaluation of definite integrals
3. Analytic functions as mappings
  3.1 The zeros of an analytic function
    3.1.1 The stability of solutions of a system of linear differential equations
  3.2 Maximum modulus and mean value
  3.3 Linear fractional transformations
  3.4 Conformal mapping
    3.4.1 Conformal mapping and flows
  3.5 The Riemann mapping theorem and Schwarz-Christoffel transformations
4. Analytic and harmonic functions in applications
  4.1 Harmonic functions
  4.2 Harmonic functions as solutions to physical problems
  4.3 Integral representations of harmonic functions
  4.4 Boundary-value problems
  4.5 Impulse functions and the Green's function of a domain
5. Transform methods
  5.1 The Fourier transform: basic properties
  5.2 Formulas Relating u and û
  5.3 The Laplace transform
  5.4 Applications of the Laplace transform to differential equations
  5.5 The Z-Transform
    5.5.1 The stability of a discrete linear system
Appendix 1. The stability of a discrete linear system
Appendix 2. A Table of Conformal Mappings
Appendix 3. A Table of Laplace Transforms
  Solutions to Odd-Numbered Exercises
  Index

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Complex Variables 2.3 out of 5 based on 0 ratings. 3 reviews.
Guest More than 1 year ago
Fisher¿s book is ideal for a first course in complex variables: the complex plane, geometry of the plane, analytic functions (zeros, singularities, residue computations), Cauchy-and residue theorems, harmonic functions, conformal mappings, boundary value problems, applications, and a lovely last chapter on transform theory, Fourier, Laplace etc, and using contour integration. Pedagogical features: The figures and illustrations are lovely! The exercises are many and well designed. Inclusion of solutions to odd-numbered exercises represents a good compromise. The book will work well for a mixed audience, students in math, in science, and in engineering alike. The presentation starts with a review of complex numbers functions and sequences, moves quickly to central aspects of complex function theory, elementary geometry, Mobius transformations, and conformal maps. The book was published first in 1990, but reprinted since by Dover, starting in 1999. It is suitable as a text or as a supplement in a beginning course in complex function theory, at the undergraduate level. And it is suitable for self-study. While it contains the standard elements in such a course, we note that a systematic treatment of physical problems comes relatively late, in Section 4.2, beginning on page 254 (a little past halfway into the book.) Some readers might want to begin with that. There are other Dover titles on the same subject, also elementary and suitable for a first course. They are slanted differently, and in particular, they point to different applications. Fisher¿s inclusion of transform theory gives this book an edge. See however also Churchill-Brown. Other Dover books: We recommend the books by Fisher, Volkovyskii et al, Silverman, Schwerdtfeger, and Flanigan all inexpensive! These books cover the fundamentals in functions of a single complex variable: analytic, harmonic, conformal mappings, and related applications. Further, there are non-Dover books such as: (a) R. V. Churchill ¿ J. W. Brown, and (b) J. E. Marsden - M. J. Hoffman both a lot more expensive. Review by Palle Jorgensen, August, 2006.
Guest More than 1 year ago
This book is terrible. All explanations are vague at best, and at worst insufficient information is given. One particular proof was so vague, it took the instructor three boards to fill in the missing information. Don't buy this book.
Guest More than 1 year ago
i have this book for class, it's diffiicult to understand. a lot of information is left out, and the author gives ambigous explinations.