This book goes for breadth rather than depth. Topics covered include fractals, recursions, solutions by radicals, Fourier and Laplace transforms, options pricing, tiling, and relativity, as well as the traditional topics of complex integration, series, and conformal mapping.
It tries to give a geometric view of everything. Mathematica is used for visualization, and to verify some of the theorems through examples. It is generally not used as an experimental or discovery tool. The amount of attention paid to Mathematica is about right: the book assumes you know in general how to use it, but points out less-known features that are especially useful for complex analysis. The complete text of the book is on the included CD-ROM as Mathematica notebooks.
This not a typical complex analysis textbook like Bak and Newman's Complex Analysis, that focuses on formulas, integrals, and series. It resembles Tristan Needham's Visual Complex Analysis in many ways. But Needham goes for depth rather than breadth: he takes a few ideas and pushes them as far as he can.
Complex Analysis with Mathematica is well done and includes many interesting things, but I'm uncertain who the audience is. The end-of-chapter exercises are routine, not challenging. There is a cookbook flavor and an emphasis on applications, which suggests it is aimed at physics and engineering students. On the other hand, it is rigorous (except for a few plausibility arguments), which suggests it is aimed at math students. But on the third hand it doesn't go deeply enough into the classical topics of complex analysis. For example, there is no analytic continuation, and no mention of the gamma or zeta functions except for one plot of each one.
Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.
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 argument 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.