Complex analysis with Mathematica

Complex Analysis with Mathematica offers a new 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, and the Mathematica code enables the user to run computer experiments.

Why you need complex numbers -- Complex algebra and geometry -- Cubics, quartics and visualization of complex roots -- Newton-Raphson iteration and complex fractals -- A complex view of the real logistic map -- The Mandelbrot set -- Symmetric chaos in the complex plane -- Complex functions -- Sequences, series and power series -- Complex differentiation -- Paths and complex integration -- Cauchy's theorem -- Cauchy's integral formula and its remarkable consequences -- Laurent series, zeroes, singularities and residues -- Residue calculus: integration, summation and the argument principle -- Conformal mapping I: simple mappings and Möbius transforms -- Fourier transforms -- Laplace transforms -- Elementary applications to two-dimensional physics -- Numerical transform techniques -- Conformal mapping II: The Schwarz-Christoffel mapping -- Tiling the Euclidean and hyperbolic planes -- Physics in three and four dimensions I -- Physics in three and four dimensions II.