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9780521836265

Complex Analysis with MATHEMATICA®

by
  • ISBN13:

    9780521836265

  • ISBN10:

    0521836263

  • Format: Hardcover
  • Copyright: 2006-05-29
  • Publisher: Cambridge University Press

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Supplemental Materials

What is included with this book?

Summary

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.

Table of 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
Why you need complex numbers
1(9)
Introduction
1(1)
First analysis of quadratic equations
1(2)
Mathematica investigation: quadratic equations
3(7)
Exercises
8(2)
Complex algebra and geometry
10(31)
Introduction
10(1)
Informal approach to 'real' numbers
10(2)
Definition of a complex number and notation
12(1)
Basic algebraic properties of complex numbers
13(1)
Complex conjugation and modulus
14(1)
The Wessel--Argand plane
14(1)
Cartesian and polar forms
15(6)
DeMoivre's theorem
21(4)
Complex roots
25(4)
The exponential form for complex numbers
29(3)
The triangle inequalities
32(1)
Mathematica visualization of complex roots and logs
33(2)
Multiplication and spacing in Mathematica
35(6)
Exercises
35(6)
Cubics, quartics and visualization of complex roots
41(15)
Introduction
41(1)
Mathematica investigation of cubic equations
42(3)
Mathematica investigation of quartic equations
45(6)
The quintic
51(1)
Root movies and root locus plots
51(5)
Exercises
53(3)
Newton--Raphson iteration and complex fractals
56(22)
Introduction
56(1)
Newton--Raphson methods
56(1)
Mathematica visualization of real Newton--Raphson
57(2)
Cayley's problem: complex global basins of attraction
59(3)
Basins of attraction for a simple cubic
62(5)
More general cubics
67(4)
Higher-order simple polynomials
71(2)
Fractal planets: Riemann sphere constructions
73(5)
Exercises
76(2)
A complex view of the real logistic map
78(27)
Introduction
78(1)
Cobwebbing theory
79(1)
Definition of the quadratic and cubic logistic maps
80(1)
The logistic map: an analytical approach
81(8)
What about n=3, 4,...?
89(2)
Summary of our root-finding investigations
91(1)
The logistic map: an experimental approach
91(1)
Experiment one: 0 < λ < 1
92(1)
Experiment two: 1 < λ < 2
93(1)
Experiment three: 2 < λ < √5
93(2)
Experiment four: 2.45044 < λ < 2.46083
95(1)
Experiment five: √5 < λ < √5 + ε
96(1)
Experiment six: √5 < λ
96(2)
Bifurcation diagrams
98(2)
Symmetry-related bifurcation
100(2)
Remarks
102(3)
Exercises
103(2)
The Mandelbrot set
105(33)
Introduction
105(1)
From the logistic map to the Mandelbrot map
105(2)
Stable fixed points: complex regions
107(3)
Periodic orbits
110(4)
Escape-time algorithm for the Mandelbrot set
114(6)
MathLink versions of the escape-time algorithm
120(6)
Diving into the Mandelbrot set: fractal movies
126(3)
Computing and drawing the Mandelbrot set
129(9)
Exercises
135(1)
Appendix: C Code listings
136(2)
Symmetric chaos in the complex plane
138(21)
Introduction
138(1)
Creating and iterating complex non-linear maps
139(4)
A movie of a symmetry-increasing bifurcation
143(2)
Visitation density plots
145(1)
High-resolution plots
146(1)
Some colour functions to try
146(2)
Hit the turbos with MathLink!
148(1)
Billion iterations picture gallery
149(10)
Exercises
154(1)
Appendix: C code listings
155(4)
Complex functions
159(35)
Introduction
159(1)
Complex functions: definitions and terminology
159(4)
Neighbourhoods, open sets and continuity
163(2)
Elementary vs. series approach to simple functions
165(4)
Simple inverse functions
169(2)
Branch points and cuts
171(4)
The Riemann sphere and infinity
175(1)
Visualization of complex functions
176(7)
Three-dimensional views of a complex function
183(4)
Holey and checkerboard plots
187(2)
Fractals everywhere?
189(5)
Exercises
192(2)
Sequences, series and power series
194(14)
Introduction
194(1)
Sequences, series and uniform convergence
194(2)
Theorems about series and tests for convergence
196(6)
Convergence of power series
202(3)
Functions defined by power series
205(1)
Visualization of series and functions
205(3)
Exercises
207(1)
Complex differentiation
208(29)
Introduction
208(1)
Complex differentiability at a point
209(2)
Real differentiability of complex functions
211(1)
Complex differentiability of complex functions
212(1)
Definition via quotient formula
213(1)
Holomorphic, analytic and regular functions
214(1)
Simple consequences of the Cauchy-Riemann equations
214(1)
Standard differentiation rules
215(2)
Polynomials and power series
217(3)
A point of notation and spotting non-analytic functions
220(1)
The Ahlfors--Struble(?) theorem
221(16)
Exercises
233(4)
Paths and complex integration
237(11)
Introduction
237(1)
Paths
237(3)
Contour integration
240(1)
The fundamental theorem of calculus
241(1)
The value and length inequalities
242(1)
Uniform convergence and integration
243(1)
Contour integration and its perils in Mathematica!
244(4)
Exercises
245(3)
Cauchy's theorem
248(15)
Introduction
248(1)
Green's theorem and the weak Cauchy theorem
248(2)
The Cauchy--Goursat theorem for a triangle
250(4)
The Cauchy--Goursat theorem for star-shaped sets
254(1)
Consequences of Cauchy's theorem
255(4)
Mathematica pictures of the triangle subdivision
259(4)
Exercises
261(2)
Cauchy's integral formula and its remarkable consequences
263(15)
Introduction
263(1)
The Cauchy integral formula
263(2)
Taylor's theorem
265(6)
The Cauchy inequalities
271(1)
Liouville's theorem
271(1)
The fundamental theorem of algebra
272(2)
Morera's theorem
274(1)
The mean-value and maximum modulus theorems
275(3)
Exercises
275(3)
Laurent series, zeroes, singularities and residues
278(24)
Introduction
278(1)
The Laurent series
278(4)
Definition of the residue
282(1)
Calculation of the Laurent series
282(4)
Definitions and properties of zeroes
286(1)
Singularities
287(5)
Computing residues
292(1)
Examples of residue computations
293(9)
Exercises
299(3)
Residue calculus: integration, summation and the argument principle
302(36)
Introduction
302(1)
The residue theorem
302(2)
Applying the residue theorem
304(1)
Trigonometric integrals
305(8)
Semicircular contours
313(3)
Semicircular contour: easy combinations of trigonometric functions and polynomials
316(2)
Mousehole contours
318(2)
Dealing with functions with branch points
320(4)
Infinitely many poles and series summation
324(4)
The argument principle and Rouche's theorem
328(10)
Exercises
335(3)
Conformal mapping I: simple mappings and Mobius transforms
338(19)
Introduction
338(1)
Recall of visualization tools
338(2)
A quick tour of mappings in Mathematica
340(7)
The conformality property
347(1)
The area-scaling property
348(1)
The fundamental family of transformations
348(1)
Group properties of the Mobius transform
349(1)
Other properties of the Mobius transform
350(4)
More about ComplexInequalityPlot
354(3)
Exercises
355(2)
Fourier transforms
357(24)
Introduction
357(1)
Definition of the Fourier transform
358(1)
An informal look at the delta-function
359(4)
Inversion, convolution, shifting and differentiation
363(3)
Jordan's lemma: semicircle theorem II
366(2)
Examples of transforms
368(4)
Expanding the setting to a fully complex picture
372(1)
Applications to differential equations
373(3)
Specialist applications and other Mathematica functions and packages
376(5)
Appendix 17: older versions of Mathematica
377(2)
Exercises
379(2)
Laplace transforms
381(20)
Introduction
381(1)
Definition of the Laplace transform
381(2)
Properties of the Laplace transform
383(4)
The Bromwich integral and inversion
387(1)
Inversion by contour integration
387(3)
Convolutions and applications to ODEs and PDEs
390(5)
Conformal maps and Efros's theorem
395(6)
Exercises
398(3)
Elementary applications to two-dimensional physics
401(32)
Introduction
401(1)
The universality of Laplace's equation
401(2)
The role of holomorphic functions
403(3)
Integral formulae for the half-plane and disk
406(2)
Fundamental solutions
408(5)
The method of images
413(2)
Further applications to fluid dynamics
415(10)
The Navier--Stokes equations and viscous flow
425(8)
Exercises
430(3)
Numerical transform techniques
433(18)
Introduction
433(1)
The discrete Fourier transform
433(2)
Applying the discrete Fourier transform in one dimension
435(2)
Applying the discrete Fourier transform in two dimensions
437(2)
Numerical methods for Laplace transform inversion
439(1)
Inversion of an elementary transform
440(1)
Two applications to 'rocket science'
441(10)
Exercises
448(3)
Conformal mapping II: the Schwarz--Christoffel mapping
451(22)
Introduction
451(1)
The Riemann mapping theorem
452(1)
The Schwarz--Christoffel transformation
452(2)
Analytical examples with two vertices
454(2)
Triangular and rectangular boundaries
456(7)
Higher-order hypergeometric mappings
463(2)
Circle mappings and regular polygons
465(5)
Detailed numerical treatments
470(3)
Exercises
470(3)
Tiling the Euclidean and hyperbolic planes
473(40)
Introduction
473(1)
Background
473(2)
Tiling the Eudlidean plane with triangles
475(6)
Tiling the Eudlidean plane with other shapes
481(4)
Triangle tilings of the Poincare disc
485(5)
Ghosts and birdies tiling of the Poincare disc
490(7)
The projective representation
497(2)
Tiling the Poincare disc with hyperbolic squares
499(8)
Heptagon tilings
507(3)
The upper half-plane representation
510(3)
Exercises
512(1)
Physics in three and four dimensions I
513(27)
Introduction
513(1)
Minkowski space and the celestial sphere
514(1)
Stereographic projection revisited
515(1)
Projective coordinates
515(2)
Mobius and Lorentz transformations
517(1)
The invisibility of the Lorentz contraction
518(2)
Outline classification of Lorentz transformations
520(4)
Warping with Mathematica
524(5)
From null directions to points: twistors
529(2)
Minimal surfaces and null curves I: holomorphic parametrizations
531(4)
Minimal surfaces and null curves II: minimal surfaces and visualization in three dimensions
535(5)
Exercises
538(2)
Physics in three and four dimensions II
540(13)
Introduction
540(1)
Laplace's equation in dimension three
540(1)
Solutions with an axial symmetry
541(2)
Translational quasi-symmetry
543(1)
From three to four dimensions and back again
544(4)
Translational symmetry: reduction to 2-D
548(2)
Comments
550(3)
Exercises
551(2)
Bibliograpy 553(5)
Index 558

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