Complex Analysis with MATHEMATICA«

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Cambridge University Press, Apr 20, 2006 - Computers - 571 pages
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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.
 

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Contents

Exercises
8
2 Complex algebra and geometry
10
22 Definition of a complex number and notation
12
24 Complex conjugation and modulus
14
27 DeMoivres theorem
21
29 The exponential form for complex numbers
29
212 Ÿ Multiplication and spacing in Mathematica
35
3 Cubics quartics and visualization of
41
Exercises
261
13 Cauchys integral formula and its
263
133 The Cauchy inequalities
271
135 The fundamental theorem of algebra
272
136 Moreras theorem
274
137 The meanvalue and maximum modulus theorems
275
14 Laurent series zeroes singularities and
278
142 Definition of the residue
282

33 The quintic
51
4 NewtonRaphson iteration and complex
56
42 Ÿ Mathematica visualization of real NewtonRaphson
57
44 Ÿ Basins of attraction for a simple cubic
62
45 Ÿ More general cubics
67
Lets look first to see what happens when r
68
46 Higherorder simple polynomials
71
Exercises
76
5 A complex view of the real logistic map
78
51 Cobwebbing theory
79
52 Ÿ Definition of the quadratic and cubic logistic maps
80
an analytical approach
81
If you are using a computer the above graphic can
88
55 Ÿ Summary of our rootfinding investigations
91
1 l 2
93
245044 l 246083
95
513 Ÿ Bifurcation diagrams
98
514 Ÿ Symmetryrelated bifurcations
100
Exercises
103
6 The Mandelbrot set
105
63 Ÿ Periodic orbits
110
Exercises
135
7 Symmetric chaos in the complex plane
138
73 Ÿ Visitation density plots
145
74 Highresolution plots
146
76 Ÿ Hit the turbos with MathLink
148
8 Complex functions
159
82 Neighbourhoods open sets and continuity
163
83 Elementary vs series approach to simple functions
165
84 Simple inverse functions
169
85 Ÿ Branch points and cuts
171
86 The Riemann sphere and infinity
175
87 Visualization of complex functions
176
89 Ÿ Holey and checkerboard plots
187
By filling in a second colour rather than a hole
188
Such routines can often make the folds and intersections of
189
Exercises
192
9 Sequences series and power series
194
92 Theorems about series and tests for convergence
196
93 Convergence of power series
202
94 Functions defined by power series
205
Exercises
207
10 Complex differentiation
208
101 Complex differentiability at a point
209
103 Complex differentiability of complex functions
212
104 Definition via quotient formula
213
105 Holomorphic analytic and regular functions
214
1010 The AhlforsStruble? theorem
221
Exercises
233
11 Paths and complex integration
237
112 Contour integration
240
The fundamental theorem of calculus
241
116 Ÿ Contour integration and its perils in Mathematica
244
12 Cauchys theorem
248
122 The CauchyGoursat theorem for a triangle
250
123 The CauchyGoursat theorem for starshaped sets
254
144 Definitions and properties of zeroes
286
integration
302
152 Applying the residue theorem
304
153 Trigonometric integrals
305
154 Semicircular contours
313
156 Mousehole contours
318
157 Dealing with functions with branch points
320
Exercises
335
simple mappings
338
162 Ÿ A quick tour of mappings in Mathematica
340
163 The conformality property
347
164 The areascaling property
348
166 Ÿ Group properties of the M÷bius transform
349
167 Other properties of the M÷bius transform
350
168 Ÿ More about ComplexInequalityPlot
354
Exercises
355
17 Fourier transforms
357
171 Definition of the Fourier transform
358
172 An informal look at the deltafunction
359
176 Expanding the setting to a fully complex picture
372
18 Laplace transforms
381
182 Properties of the Laplace transform
383
183 The Bromwich integral and inversion
387
185 Convolutions and applications to ODEs and PDEs
390
186 Conformal maps and Efross theorem
395
19 Elementary applications to
401
192 The role of holomorphic functions
403
193 Integral formulae for the halfplane and disk
406
195 The method of images
413
20 Numerical transform techniques
433
202 Ÿ Applying the discrete Fourier transform in one
435
203 Ÿ Applying the discrete Fourier transform in two
437
205 Ÿ Inversion of an elementary transform
440
the
451
211 The Riemann mapping theorem
452
215 Ÿ Higherorder hypergeometric mappings
463
217 Detailed numerical treatments
470
22 Tiling the Euclidean and hyperbolic
473
222 Ÿ Tiling the Euclidean plane with triangles
475
a
476
a
477
223 Ÿ Tiling the Euclidean plane with other shapes
481
224 Ÿ Triangle tilings of the PoincarÚ disc
485
For larger values of n the corresponding projective picture approaches
504
n 14
505
23 Physics in three and four dimensions I
513
231 Minkowski space and the celestial sphere
514
232 Stereographic projection revisited
515
234 M÷bius and Lorentz transformations
517
237 Warping with Mathematica
524
twistors
529
holomorphic
531
minimal
535
24 Physics in three and four dimensions II
540
243 Translational quasisymmetry
543

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About the author (2006)

William Shaw is a tutor in Mathematics at St Catherine's College, Oxford.

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