## Complex Analysis with MATHEMATICA®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 Möbius transform | 349 |

167 Other properties of the Möbius 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 Poincaré 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 Möbius 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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