Complex analysis with Mathematica:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Ausgabe: | 1. publ. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXV, 571 S. Ill., graph. Darst. CD-ROM (12 cm) |
| ISBN: | 0521836263 9780521836265 |
Internformat
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| 245 | 1 | 0 | |a Complex analysis with Mathematica |c William T. Shaw |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2006 | |
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Datensatz im Suchindex
| _version_ | 1804133322231644160 |
|---|---|
| adam_text | COMPLEX ANALYSIS WITH MATHEMATICA WILLIAM T. SHAW ST CATHERINE S
COLLEGE, OXFORD AND OXFORD CENTRE FOR INDUSTRIAL AND APPLIED MATHEMATICS
CAMBRIDGE UNIVERSITY PRESS 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 1 1.1 FIRST
ANALYSIS OF QUADRATIC EQUATIONS 1 1.2 MATHEMATICA INVESTIGATION:
QUADRATIC EQUATIONS 3 EXERCISES , 8 2 COMPLEX ALGEBRA AND GEOMETRY 10
INTRODUCTION 10 2.1 INFORMAL APPROACH TO REAL NUMBERS 10 2.2
DEFINITION OF A COMPLEX NUMBER AND NOTATION 12 2.3 BASIC ALGEBRAIC
PROPERTIES OF COMPLEX NUMBERS 13 2.4 COMPLEX CONJUGATION AND MODULUS 14
2.5 THE WESSEL-ARGAND PLANE 14 2.6 CARTESIAN AND POLAR FORMS 15 2.7
DEMOIVRE S THEOREM 21 2.8 COMPLEX ROOTS 25 2.9 THE EXPONENTIAL FORM FOR
COMPLEX NUMBERS 29 2.10 THE TRIANGLE INEQUALITIES 32 2.11 MATHEMATICA
VISUALIZATION OF COMPLEX ROOTS AND LOGS 33 2.12 MULTIPLICATION AND
SPACING IN MATHEMATICA 35 EXERCISES ^ 35 3 CUBICS, QUARTICS AND
VISUALIZATION OF COMPLEX ROOTS 41 INTRODUCTION 41 3.1 MATHEMATICA
INVESTIGATION OF CUBIC EQUATIONS 42 3.2 MATHEMATICA INVESTIGATION OF
QUARTIC EQUATIONS 45 VIII CONTENTS 3.3 THE QUINTIC 51 3.4 ROOT MOVIES
AND ROOT LOCUS PLOTS 51 EXERCISES . 5 3 4 NEWTON*RAPHSON ITERATION AND
COMPLEX FRACTALS 56 INTRODUCTION 56 4.1 NEWTON-RAPHSON METHODS 56 4.2
MATHEMATICA VISUALIZATION OF REAL NEWTON-RAPHSON 57 4.3 CAYLEY S
PROBLEM: COMPLEX GLOBAL BASINS OF ATTRACTION 59 4.4 BASINS OF ATTRACTION
FOR A SIMPLE CUBIC 62 4.5 MORE GENERAL CUBICS 67 4.6 HIGHER-ORDER SIMPLE
POLYNOMIALS 71 4.7 FRACTAL PLANETS: RIEMANN SPHERE CONSTRUCTIONS 73
EXERCISES 76 5 A COMPLEX VIEW OF THE REAL LOGISTIC MAP 78 INTRODUCTION
78 5.1 COBWEBBING THEORY 79 5.2 DEFINITION OF THE QUADRATIC AND CUBIC
LOGISTIC MAPS 80 5.3 THE LOGISTIC MAP: AN ANALYTICAL APPROACH 81 5.4
WHAT ABOUT N=3,4,...? 89 5.5 SUMMARY OF OUR ROOT-FINDING INVESTIGATIONS
91 5.6 THE LOGISTIC MAP: AN EXPERIMENTAL APPROACH 91 5.7 EXPERIMENT ONE:
0 A 1 92 5.8 EXPERIMENT TWO: 1 A 2 93 5.9 EXPERIMENT THREE: 2
A Y/5 93 5.10 EXPERIMENT FOUR: 2.45044 A 2.46083 95 5.11
EXPERIMENT FIVE: Y/B A Y/Z + E 96 5.12 EXPERIMENT SIX: /5 A 96
5.13 BIFURCATION DIAGRAMS 98 5.14 SYMMETRY-RELATED BIFURCATION * 100
5.15 REMARKS 102 EXERCISES 103 6 THE MANDELBROT SET 105 INTRODUCTION 105
6.1 FROM THE LOGISTIC MAP TO THE MANDELBROT MAP 105 6.2 - STABLE FIXED
POINTS: COMPLEX REGIONS 107 6.3 PERIODIC ORBITS 110 6.4 ESCAPE-TIME
ALGORITHM FOR THE MANDELBROT SET 114 6.5 MATHLINK VERSIONS OF THE
ESCAPE-TIME ALGORITHM 120 6.6 DIVING INTO THE MANDELBROT SET: FRACTAL
MOVIES 126 6.7 COMPUTING AND DRAWING THE MANDELBROT SET 129 EXERCISES .
135 APPENDIX: C CODE LISTINGS 136 CONTENTS IX 7 SYMMETRIC CHAOS IN THE
COMPLEX PLANE 138 INTRODUCTION _ 138 7.1 CREATING AND ITERATING COMPLEX
NON-LINEAR MAPS 139 7.2 A MOVIE OF A SYMMETRY-INCREASING BIFURCATION -,
143 7.3 VISITATION DENSITY PLOTS 145 7.4 HIGH-RESOLUTION PLOTS . 146 7.5
SOME COLOUR FUNCTIONS TO TRY 146 7.6 HIT THE TURBOS WITH MATHLINKL 148
7.7 BILLION ITERATIONS PICTURE GALLERY 149 EXERCISES *, 154 APPENDIX: C
CODE LISTINGS * 155 8 COMPLEX FUNCTIONS 159 INTRODUCTION 159 8.1 COMPLEX
FUNCTIONS: DEFINITIONS AND TERMINOLOGY 159 8.2 NEIGHBOURHOODS, OPEN SETS
AND CONTINUITY 163 8.3 ELEMENTARY VS. SERIES APPROACH TO SIMPLE
FUNCTIONS 165 8.4 SIMPLE INVERSE FUNCTIONS 169 8.5 BRANCH POINTS AND
CUTS 171 8.6 THE RIEMANN SPHERE AND INFINITY 175 8.7 VISUALIZATION OF
COMPLEX FUNCTIONS 176 8.8 THREE-DIMENSIONAL VIEWS OF A COMPLEX FUNCTION
183 8.9 HOLEY AND CHECKERBOARD PLOTS 187 8.10 FRACTALS EVERYWHERE? 189
EXERCISES * * 192 9 SEQUENCES, SERIES AND POWER SERIES 194 INTRODUCTION
194 9.1 SEQUENCES, SERIES AND UNIFORM CONVERGENCE 194 9.2 THEOREMS ABOUT
SERIES AND TESTS FOR CONVERGENCE 196 9.3 CONVERGENCE OF POWER SERIES 202
9.4 FUNCTIONS DEFINED BY POWER SERIES 205 9.5 VISUALIZATION OF SERIES
AND FUNCTIONS 205 EXERCISES 207 10 COMPLEX DIFFERENTIATION 208
INTRODUCTION 208 10.1 COMPLEX DIFFERENTIABILITY AT A POINT 209 10.2 REAL
DIFFERENTIABILITY OF COMPLEX FUNCTIONS 211 10.3 COMPLEX
DIFFERENTIABILITY OF COMPLEX FUNCTIONS 212 10.4 DEFINITION VIA QUOTIENT
FORMULA 213 10.5 HOLOMORPHIC, ANALYTIC AND REGULAR FUNCTIONS 214 10.6
SIMPLE CONSEQUENCES OF THE CAUCHY-RIEMANN EQUATIONS 214 10.7 STANDARD
DIFFERENTIATION RULES 215 10.8 POLYNOMIALS AND POWER SERIES 217 10.9 A
POINT OF NOTATION AND SPOTTING NON-ANALYTIC FUNCTIONS 220 CONTENTS 10.10
THE AHLFORS-STRUBLE(?) THEOREM 221 EXERCISES 233 11 PATHS AND COMPLEX
INTEGRATION 237 INTRODUCTION 237 11.1 PATHS 237 11.2 CONTOUR
INTEGRATION 240 11.3 THE FUNDAMENTAL THEOREM OF CALCULUS 241 11.4 THE
VALUE AND LENGTH INEQUALITIES 242 11.5 UNIFORM CONVERGENCE AND
INTEGRATION , 243 11.6 CONTOUR INTEGRATION AND ITS PERILS IN
MATHEMATICA! 244 EXERCISES 245 12 CAUCHY S THEOREM 248 INTRODUCTION 248
12.1 GREEN S THEOREM AND THE WEAK CAUCHY THEOREM 248 12.2 THE
CAUCHY-GOURSAT THEOREM FOR A TRIANGLE 250 12.3 THE CAUCHY-GOURSAT
THEOREM FOR STAR-SHAPED SETS 254 12.4 CONSEQUENCES OF CAUCHY S THEOREM
255 12.5 MATHEMATICA PICTURES OF THE TRIANGLE SUBDIVISION 259 EXERCISES
261 13 CAUCHY S INTEGRAL FORMULA AND ITS REMARKABLE CONSEQUENCES 263
INTRODUCTION 263 13.1 THE CAUCHY INTEGRAL FORMULA 263 13.2 TAYLOR S
THEOREM 265 13.3 THE CAUCHY INEQUALITIES 271 13.4 LIOUVILLE S THEOREM
271 13.5 THE FUNDAMENTAL THEOREM OF ALGEBRA 272 13.6 MORERA S THEOREM
274 13.7 THE MEAN-VALUE AND MAXIMUM MODULUS THEOREMS 275 EXERCISES 275
14 LAURENT SERIES, ZEROES, SINGULARITIES AND RESIDUES 278 INTRODUCTION
278 14.1 THE LAURENT SERIES 278 14.2 DEFINITION OF THE RESIDUE 282 14.3
CALCULATION OF THE LAURENT SERIES 282 14.4 DEFINITIONS AND PROPERTIES OF
ZEROES 286 14.5 SINGULARITIES 287 14.6 COMPUTING RESIDUES 292 14.7
EXAMPLES OF RESIDUE COMPUTATIONS 293 EXERCISES 299 CONTENTS . * . XI 15
RESIDUE CALCULUS: INTEGRATION, SUMMATION AND THE ARGUMENT PRINCIPLE 302
INTRODUCTION 302 15.1 THE RESIDUE THEOREM 302 15.2 APPLYING THE RESIDUE
THEOREM * 304 15.3 TRIGONOMETRIC INTEGRALS 305 15.4 SEMICIRCULAR
CONTOURS F 313 15.5 SEMICIRCULAR CONTOUR: EASY COMBINATIONS OF
TRIGONOMETRIC FUNCTIONS AND POLYNOMIALS 316 15.6 MOUSEHOLE CONTOURS 318
15.7 DEALING WITH FUNCTIONS WITH BRANCH POINTS 320 15.8 INFINITELY MANY
POLES AND SERIES SUMMATION 324 15.9 THE ARGUMENT PRINCIPLE AND ROUCHE S
THEOREM 328 EXERCISES 335 16 CONFORMAL MAPPING I: SIMPLE MAPPINGS AND
MOBIUS TRANS- FORMS 338 INTRODUCTION 338 16.1 RECALL OF VISUALIZATION
TOOLS 338 16.2 A QUICK TOUR OF MAPPINGS IN MATHEMATICA 340 16.3 THE
CONFORMALITY PROPERTY 347 16.4 THE AREA-SCALING PROPERTY 348 16.5 THE
FUNDAMENTAL FAMILY OF TRANSFORMATIONS 348 16.6 GROUP PROPERTIES OF THE
MOBIUS TRANSFORM 349 16.7 OTHER PROPERTIES OF THE MOBIUS TRANSFORM 350
16.8 MORE ABOUT COMPLEXINEQUALITYPLOT 354 EXERCISES 355 17 FOURIER
TRANSFORMS 357 INTRODUCTION 357 17.1 DEFINITION OF THE FOURIER TRANSFORM
358 17.2 AN INFORMAL LOOK AT THE DELTA-FUNCTION 359 17.3 INVERSION,
CONVOLUTION, SHIFTING AND DIFFERENTIATION 363 17.4 JORDAN S LEMMA:
SEMICIRCLE THEOREM II 366 17.5 EXAMPLES OF TRANSFORMS . 368 17.6
EXPANDING THE SETTING TO A FULLY COMPLEX PICTURE 372 17.7 APPLICATIONS
TO DIFFERENTIAL EQUATIONS 373 17.8 SPECIALIST APPLICATIONS AND OTHER
MATHEMATICA FUNCTIONS AND PACKAGES 376 APPENDIX 17: OLDER VERSIONS OF
MATHEMATICA 377 EXERCISES 379 18 LAPLACE TRANSFORMS 381 INTRODUCTION 381
18.1 DEFINITION OF THE LAPLACE TRANSFORM 381 18.2 PROPERTIES OF THE
LAPLACE TRANSFORM 383 XII CONTENTS 18.3 THE BROMWICH INTEGRAL AND
INVERSION 387 18.4 INVERSION BY CONTOUR INTEGRATION 387 18.5
CONVOLUTIONS AND APPLICATIONS TO ODES AND PDES 390 18.6 CONFORMAL MAPS
AND EFROS S THEOREM 395 EXERCISES 398 19 ELEMENTARY APPLICATIONS TO
TWO-DIMENSIONAL PHYSICS 401 INTRODUCTION 401 19.1 THE UNIVERSALITY OF
LAPLACE S EQUATION 401 19.2 THE ROLE OF HOLOMORPHIC FUNCTIONS 403 19.3
INTEGRAL FORMULAE FOR THE HALF-PLANE AND DISK 406 19.4 FUNDAMENTAL
SOLUTIONS 408 19.5 THE METHOD OF IMAGES 413 19.6 FURTHER APPLICATIONS TO
FLUID DYNAMICS 415 19.7 THE NAVIER-STOKES EQUATIONS AND VISCOUS FLOW 425
EXERCISES 430 20 NUMERICAL TRANSFORM TECHNIQUES 433 INTRODUCTION 433
20.1 THE DISCRETE FOURIER TRANSFORM 433 20.2 APPLYING THE DISCRETE
FOURIER TRANSFORM IN ONE DIMENSION 435 20.3 APPLYING THE DISCRETE
FOURIER TRANSFORM IN TWO DIMENSIONS 437 20.4 NUMERICAL METHODS FOR
LAPLACE TRANSFORM INVERSION 439 20.5 INVERSION OF AN ELEMENTARY
TRANSFORM . 440 20.6 TWO APPLICATIONS TO ROCKET SCIENCE 441 EXERCISES
448 21 CONFORMAL MAPPING II: THE SCHWARZ*CHRISTOFFEL MAPPING 451
INTRODUCTION 451 21.1 THE RIEMANN MAPPING THEOREM 452 21.2 THE
SCHWARZ-CHRISTOFFEL TRANSFORMATION 452 21.3 ANALYTICAL EXAMPLES WITH TWO
VERTICES 454 21.4 TRIANGULAR AND RECTANGULAR BOUNDARIES 456 21.5
HIGHER-ORDER HYPERGEOMETRIC MAPPINGS 463 21.6 CIRCLE MAPPINGS AND
REGULAR POLYGONS 465 21.7 DETAILED NUMERICAL TREATMENTS 470 EXERCISES
470 22 TILING THE EUCLIDEAN AND HYPERBOLIC PLANES 473 INTRODUCTION 473
22.1 BACKGROUND 473 22.2 TILING THE EUDLIDEAN PLANE WITH TRIANGLES 475
22.3 TILING THE EUDLIDEAN PLANE WITH OTHER SHAPES 481 22.4 TRIANGLE
TILINGS OF THE POINCARE DISC 485 22.5 GHOSTS AND BIRDIES TILING OF THE
POINCARE DISC 490 22.6 THE PROJECTIVE REPRESENTATION 497 CONTENTS XIII
22.7 TILING THE POINCARE DISC WITH HYPERBOLIC SQUARES 499 22.8 HEPTAGON
TILINGS 507 22.9 THE UPPER HALF-PLANE REPRESENTATION 510 EXERCISES *-
512 23 PHYSICS IN THREE AND FOUR DIMENSIONS I 513 INTRODUCTION 513 23.1
MINKOWSKI SPACE AND THE CELESTIAL SPHERE 514 23.2 STEREOGRAPHIC
PROJECTION REVISITED 515 23.3 PROJECTIVE COORDINATES 515 23.4 MOBIUS AND
LORENTZ TRANSFORMATIONS 517 23.5 THE INVISIBILITY OF THE LORENTZ
CONTRACTION 518 23.6 OUTLINE CLASSIFICATION OF LORENTZ TRANSFORMATIONS
520 23.7 WARPING.WITH MATHEMATICA 524 23.8 FROM NULL DIRECTIONS TO
POINTS: TWISTORS 529 23.9 MINIMAL SURFACES AND NULL CURVES I:
HOLOMORPHIC PARAMETRIZA- TIONS * 531 23.10 MINIMAL SURFACES AND NULL
CURVES II: MINIMAL SURFACES AND VISUALIZATION IN THREE DIMENSIONS 535
EXERCISES 538 24 PHYSICS IN THREE AND FOUR DIMENSIONS II 540
INTRODUCTION 540 24.1 LAPLACE S EQUATION IN DIMENSION THREE 540 24.2
SOLUTIONS WITH AN AXIAL SYMMETRY 541 24.3 TRANSLATIONAL QUASI-SYMMETRY
543 24.4 FROM THREE TO FOUR DIMENSIONS AND BACK AGAIN 544 24.5
TRANSLATIONAL SYMMETRY: REDUCTION TO 2-D 548 24.6 COMMENTS 550 EXERCISES
551 BIBLIOGRAPY . 553 INDEX 558
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T20:06:52Z |
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| isbn | 0521836263 9780521836265 |
| language | English |
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| owner_facet | DE-20 DE-824 |
| physical | XXV, 571 S. Ill., graph. Darst. CD-ROM (12 cm) |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Cambridge Univ. Press |
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| spelling | Shaw, William T. Verfasser aut Complex analysis with Mathematica William T. Shaw 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2006 XXV, 571 S. Ill., graph. Darst. CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Mathematica (Computer file) Funktionentheorie - Mathematica <Programm> Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Mathematica Programm (DE-588)4268208-3 gnd rswk-swf Funktionentheorie (DE-588)4018935-1 s Mathematica Programm (DE-588)4268208-3 s DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013145212&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shaw, William T. Complex analysis with Mathematica Mathematica (Computer file) Funktionentheorie - Mathematica <Programm> Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Mathematica Programm (DE-588)4268208-3 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4268208-3 |
| title | Complex analysis with Mathematica |
| title_auth | Complex analysis with Mathematica |
| title_exact_search | Complex analysis with Mathematica |
| title_full | Complex analysis with Mathematica William T. Shaw |
| title_fullStr | Complex analysis with Mathematica William T. Shaw |
| title_full_unstemmed | Complex analysis with Mathematica William T. Shaw |
| title_short | Complex analysis with Mathematica |
| title_sort | complex analysis with mathematica |
| topic | Mathematica (Computer file) Funktionentheorie - Mathematica <Programm> Functions of complex variables Mathematical analysis Funktionentheorie (DE-588)4018935-1 gnd Mathematica Programm (DE-588)4268208-3 gnd |
| topic_facet | Mathematica (Computer file) Funktionentheorie - Mathematica <Programm> Functions of complex variables Mathematical analysis Funktionentheorie Mathematica Programm |
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