








Complex Analysis with Mathematica






Organization:  King's College, University of London 
Department:  Department of Mathematics 






Publisher:  Cambridge University Press (United Kingdom) 
  






Why You Need Complex Numbers  Complex Algebra and Geometry  Cubics, Quartics and Visualization of Complex Roots  NewtonRaphson 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 TwoDimensional Physics  Numerical Transform Techniques  Conformal Mapping II: The SchwarzChristoffel Mapping  Tiling the Euclidean and Hyperbolic Planes  Physics in Three and Four Dimensions I  Physics in Three and Four Dimensions II  Bibliography






This book presents complex numbers in a stateoftheart computational environment. Its innovative approach also offers insights into areas too often neglected in a student treatment, including complex chaos, mathematical art, physics in three or more dimensions, and advanced fluid dynamics. Integration with Mathematica allows topics not usually presentable on a blackboard, such as iterative equationsolving, as well as full graphical exploration of all areas covered. The included CD contains a live version of the book with all of the Mathematica code, allowing users to run computer experiments. Teachers can utilize the book for a traditional course, with Mathematica as a tool for illustration or for checking. Readers will enjoy it for selfstudy and enrichment. Downloads, tips, reviews, and other information are available at the book's web page. Additional Mathematica 6 notebooks to accompany this book are available in the Wolfram Library Archive.












complex numbers, complex algebra, complex geometry, WesselArgand plane, DeMoivre's theorem, cubics, quartics, complex roots, root locus plots, root movies, quintic, Cayley's problem, Cobwebbing theory, bifurcation diagrams, symmetryrelated bifurcation, logistic map, Mandelbrot map, stable fixed points, Escapetime algorithm, nonlinear maps, Visitation denisty plots, Riemann sphere, Holey plots, checkerboard plots, complex differentiability, holomorphic functions, analytic functions, regular functions, CauchyRiemann equations, AhlforsStruble theorem, CauchyGoursat theorem, Liouville's theorem, Morera's theorem, Laurent series, residue theorem, mousehole contours, argument principle, Rouche's theorem, Mobius transform, Jordan's lemma, Bromwich integral, Bromwich inversion, Efros's theorem, NavierStokes equations, SchwarzChristoffel transformation, Poincare disk, Minkowski space, heptagon tilings













     
