Complex Analysis with MATHEMATICA®

by William T. Shaw

• ISBN13: 9780521836265
• ISBN10: 0521836263
• Format: Hardcover
• Copyright: 2006-05-29
• Publisher: Cambridge University Press

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