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Summary
Summary
Complex Analysis with Mathematica offers a way of learning and teaching a subject that lies at the heart of many areas of pure and applied mathematics, physics, engineering and even art. This book offers teachers and students an opportunity to learn about complex numbers in a state-of-the-art computational environment. The innovative approach also offers insights into many areas too often neglected in a student treatment, including complex chaos and mathematical art. Thus readers can also use the book for self-study and for enrichment. The use of Mathematica enables the author to cover several topics that are often absent from a traditional treatment. Students are also led, optionally, into cubic or quartic equations, investigations of symmetric chaos and advanced conformal mapping. A CD is included which contains a live version of the book: in particular all the Mathematica code enables the user to run computer experiments.
Reviews 1
Choice Review
Traditionally, teaching complex analysis requires overcoming steep pedagogical hurdles. Complex analytic functions, the objects of study, have graphs that form two-dimensional subsets of four-dimensional space, so their visualization (if it happens at all) requires indirect approaches--the more the better. Also, many important results in complex analysis (conformal mapping) express themselves in only very limited examples such as one may work out by hand. So computers should transform the teaching of the subject, as Tristan Needham pointed out in his breakthrough 1997 Visual Complex Analysis (CH, Jan'98, 35-2767). But Needham avoided integrating computer technology into his exposition, presumably because of limited distribution of powerful hardware and awkwardness of then-available software. Shaw (St. Catherine's College, Oxford), taking advantage of considerable technological progress, offers a truly next-generation book: Mathematica's graphics make old ideas leap off the page, and Mathematica's computational powers permit students to interact effectively with complicated and typical examples. Technology aside, Shaw expands typical undergraduate complex analysis itinerary with excursions through dynamics and fractals, solution of cubic and quartic equations, hyperbolic tessellations, and even an introduction to spinors and twistors, important tools in contemporary approaches to relativity. A vital library resource. Summing Up: Essential. General readers; lower- and upper-division undergraduates; professionals; two-year technical program students. D. V. Feldman University of New Hampshire
Table of Contents
| Preface |
| 1 Why you need complex numbers |
| 2 Complex algebra and geometry |
| 3 Cubics, quartics and visualization of complex roots |
| 4 Newton-Raphson iteration and complex fractals |
| 5 A complex view of the real logistic map |
| 6 The Mandelbrot set |
| 7 Symmetric chaos in the complex plane |
| 8 Complex functions |
| 9 Sequences, series and power series |
| 10 Complex differentiation |
| 11 Paths and complex integration |
| 12 Cauchy's theorem |
| 13 Cauchy's integral formula and its remarkable consequences |
| 14 Laurent series, zeroes, singularities and residues |
| 15 Residue calculus: integration, summation and the augment principle |
| 16 Conformal mapping I: simple mappings and Mobius transforms |
| 17 Fourier transforms |
| 18 Laplace transforms |
| 19 Elementary applications to two-dimensional physics |
| 20 Numerical transform techniques |
| 21 Conformal mapping II: the Schwarz-Christoffel transformation |
| 22 Tiling the Euclidean and hyperbolic planes |
| 23 Physics in three and four dimensions I |
| 24 Physics in three and four dimensions II |
| Index |
