Complex Analysis with MATHEMATICA

Transcription

1 Complex Analysis with MATHEMATICA William T. Shaw St Catherine's College, Oxford and Oxford Centre for Industrial and Applied Mathematics CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface xv Why this book? xv How this text is organized xvi Some suggestions on how to use this text xxi About the enclosed CD xxii Exercises and solutions xxiv Acknowledgements. xxiv 1 Why you need complex numbers 1 Introduction First analysis of quadratic equations Mathematica investigation: quadratic equations 3 Exercises ', 8 2 Complex algebra and geometry 10 Introduction Informal approach to 'real' numbers Definition of a complex number and notation Basic algebraic properties of complex numbers Complex conjugation and modulus The Wessel-Argand plane Cartesian and polar forms DeMoivre's theorem Complex roots The exponential form for complex numbers The triangle inequalities Mathematica visualization of complex roots and logs Multiplication and spacing in Mathematica 35 Exercises ^ 35 3 Cubics, quartics and visualization of complex roots 41 Introduction Mathematica investigation of cubic equations Mathematica investigation of quartic equations 45

3 viii Contents 3.3 The quintic Root movies and root locus plots 51 Exercises Newton Raphson iteration and complex fractals 56 Introduction Newton-Raphson methods Mathematica visualization of real Newton-Raphson Cayley's problem: complex global basins of attraction Basins of attraction for a simple cubic More general cubics Higher-order simple polynomials Fractal planets: Riemann sphere constructions 73 Exercises 76 5 A complex view of the real logistic map 78 Introduction Cobwebbing theory Definition of the quadratic and cubic logistic maps The logistic map: an analytical approach What about n=3,4,...? Summary of our root-finding investigations The logistic map: an experimental approach Experiment one: 0 < A < Experiment two: 1 < A < Experiment three: 2 < A < y/ Experiment four: < A < Experiment five: y/b < A < y/z + e Experiment six: \/5 < A Bifurcation diagrams Symmetry-related bifurcation Remarks 102 Exercises The Mandelbrot set 105 Introduction From the logistic map to the Mandelbrot map Stable fixed points: complex regions Periodic orbits Escape-time algorithm for the Mandelbrot set MathLink versions of the escape-time algorithm Diving into the Mandelbrot set: fractal movies Computing and drawing the Mandelbrot set 129 Exercises. ' 135 Appendix: C Code listings 136

4 Contents ix 7 Symmetric chaos in the complex plane 138 Introduction _ Creating and iterating complex non-linear maps A movie of a symmetry-increasing bifurcation -, Visitation density plots High-resolution plots Some colour functions to try Hit the turbos with MathLinkl Billion iterations picture gallery 149 Exercises, 154 Appendix: C code listings Complex functions 159 Introduction Complex functions: definitions and terminology Neighbourhoods, open sets and continuity Elementary vs. series approach to simple functions Simple inverse functions Branch points and cuts The Riemann sphere and infinity Visualization of complex functions Three-dimensional views of a complex function Holey and checkerboard plots Fractals everywhere? 189 Exercises * Sequences, series and power series 194 Introduction Sequences, series and uniform convergence Theorems about series and tests for convergence Convergence of power series Functions defined by power series Visualization of series and functions 205 Exercises Complex differentiation 208 Introduction Complex differentiability at a point Real differentiability of complex functions Complex differentiability of complex functions Definition via quotient formula Holomorphic, analytic and regular functions Simple consequences of the Cauchy-Riemann equations Standard differentiation rules Polynomials and power series A point of notation and spotting non-analytic functions 220

5 Contents The Ahlfors-Struble(?) theorem 221 Exercises Paths and complex integration " 237 Introduction Paths Contour integration The fundamental theorem of calculus The value and length inequalities Uniform convergence and integration, Contour integration and its perils in Mathematica! 244 Exercises Cauchy's theorem 248 Introduction Green's theorem and the weak Cauchy theorem The Cauchy-Goursat theorem for a triangle The Cauchy-Goursat theorem for star-shaped sets Consequences of Cauchy's theorem Mathematica pictures of the triangle subdivision 259 Exercises Cauchy's integral formula and its remarkable consequences 263 Introduction The Cauchy integral formula Taylor's theorem The Cauchy inequalities Liouville's theorem The fundamental theorem of algebra Morera's theorem ' The mean-value and maximum modulus theorems 275 Exercises Laurent series, zeroes, singularities and residues 278 Introduction The Laurent series Definition of the residue Calculation of the Laurent series Definitions and properties of zeroes "Singularities Computing residues Examples of residue computations 293 Exercises 299

6 Contents.. xi 15 Residue calculus: integration, summation and the argument principle 302 Introduction The residue theorem Applying the residue theorem > Trigonometric integrals Semicircular contours f Semicircular contour: easy combinations of trigonometric functions and polynomials Mousehole contours Dealing with functions with branch points Infinitely many poles and series summation The argument principle and Rouche's theorem 328 Exercises Conformal mapping I: simple mappings and Mobius transforms 338 Introduction Recall of visualization tools A quick tour of mappings in Mathematica The conformality property The area-scaling property The fundamental family of transformations Group properties of the Mobius transform Other properties of the Mobius transform More about ComplexInequalityPlot 354 Exercises Fourier transforms 357 Introduction Definition of the Fourier transform An informal look at the delta-function Inversion, convolution, shifting and differentiation Jordan's lemma: semicircle theorem II Examples of transforms Expanding the setting to a fully complex picture Applications to differential equations Specialist applications and other Mathematica functions and packages 376 Appendix 17: older versions of Mathematica 377 Exercises Laplace transforms 381 Introduction Definition of the Laplace transform Properties of the Laplace transform 383

7 xii Contents 18.3 The Bromwich integral and inversion Inversion by contour integration Convolutions and applications to ODEs and PDEs Conformal maps and Efros's theorem 395 Exercises Elementary applications to two-dimensional physics 401 Introduction The universality of Laplace's equation The role of holomorphic functions Integral formulae for the half-plane and disk Fundamental solutions The method of images Further applications to fluid dynamics The Navier-Stokes equations and viscous flow 425 Exercises Numerical transform techniques 433 Introduction The discrete Fourier transform Applying the discrete Fourier transform in one dimension Applying the discrete Fourier transform in two dimensions Numerical methods for Laplace transform inversion Inversion of an elementary transform Two applications to 'rocket science' 441 Exercises Conformal mapping II: the Schwarz Christoffel mapping 451 Introduction The Riemann mapping theorem The Schwarz-Christoffel transformation Analytical examples with two vertices " Triangular and rectangular boundaries Higher-order hypergeometric mappings Circle mappings and regular polygons Detailed numerical treatments 470 Exercises Tiling the Euclidean and hyperbolic planes 473 Introduction Background Tiling the Eudlidean plane with triangles Tiling the Eudlidean plane with other shapes Triangle tilings of the Poincare disc Ghosts and birdies tiling of the Poincare disc The projective representation 497

8 Contents xiii 22.7 Tiling the Poincare disc with hyperbolic squares Heptagon tilings The upper half-plane representation 510 Exercises Physics in three and four dimensions I 513 Introduction Minkowski space and the celestial sphere Stereographic projection revisited Projective coordinates Mobius and Lorentz transformations The invisibility of the Lorentz contraction Outline classification of Lorentz transformations Warping.with Mathematica From null directions to points: twistors Minimal surfaces and null curves I: holomorphic parametrizations * Minimal surfaces and null curves II: minimal surfaces and visualization in three dimensions " 535 Exercises Physics in three and four dimensions II 540 Introduction Laplace's equation in dimension three Solutions with an axial symmetry Translational quasi-symmetry From three to four dimensions and back again Translational symmetry: reduction to 2-D Comments 550 Exercises 551 Bibliograpy. 553 Index 558