Tristan Needham9780198534471, 0198534477
Table of contents :
Preface……Page 5
Acknowledgements……Page 10
Contents……Page 12
Historical Skentch……Page 21
Bombelli´s “Wild Thought”……Page 23
Some terminology and notation……Page 26
Practice……Page 27
Equivalence of Symbolic and geometric arithmetic……Page 28
Introduction……Page 30
Moving particle argument……Page 31
Power series argument……Page 32
Introduction……Page 34
Trigonometry……Page 35
Geometry……Page 36
Calculus……Page 40
Algebra……Page 42
Vectorial operations……Page 47
Geometry through the eyes of Felix Klein……Page 50
Classifying motions……Page 54
Three reflections theorem……Page 57
Similarities and Complex arithmetic……Page 59
Spatial complex numers?……Page 63
Excercises……Page 65
Introduction……Page 75
Positive Integer Powers……Page 77
Cubics revisited *……Page 79
Cassinian Curves *……Page 80
The mystery of real power series……Page 84
The disc of convergence……Page 87
Approximating a power series with a polynomial……Page 90
Uniqueness……Page 91
Manipulating power series……Page 92
Finding the radius of convergence……Page 94
Fourier series*……Page 97
Power series approach……Page 99
The geometry of the mapping……Page 100
Another approach……Page 101
Definitions and identities……Page 104
Relation to hyperbolic functions……Page 106
The geometry of the mapping……Page 108
Example: Fractional powers……Page 110
Single-valued branches of a multifunction……Page 112
Relevance to power series……Page 115
An example with two branch points……Page 116
Inverse of the exponential function……Page 118
The logarithmic power series……Page 120
General powers……Page 121
The centroid……Page 122
Averaging over regular polygons……Page 125
Averaging over circles……Page 128
Exercises……Page 131
Connection with Einstein´s theory of relativity*……Page 142
Preliminary definitions and facts……Page 144
Preservation of circles……Page 146
Construction using orthogonal circles……Page 149
Preservation of angles……Page 150
Inversion in a sphere……Page 153
A problem on touching circles……Page 156
Quadrilaterals with orthogonal diagonals……Page 157
Ptolemy´s theorem……Page 158
The point at infinity……Page 159
Stereografic projection……Page 160
Transferring complex functions to the sphere……Page 163
Behaviour of functions at infinity……Page 164
Stereographic formulae……Page 166
Preservation of circles, angles and symmetry……Page 168
Non-uniqueness of the coefficients……Page 169
The group property……Page 170
Fixed points……Page 171
Fixed points at infinity……Page 172
The cross-ratio……Page 174
Evidence of a link with linear algebra……Page 176
The explanation: Homogeneous coordinates……Page 177
Eigenvectors and eigenvalues……Page 179
Rotations of the sphere……Page 181
The main idea……Page 182
Elliptic, hiperbolic, and loxodromic types……Page 184
Local geometric inerpretation of the multipler……Page 186
Parabolic transformations……Page 188
Computing the multipler……Page 189
Eingenvalue interpretation of the multipler……Page 190
Elliptic case……Page 192
Hyperbolic case……Page 193
Parabolic case……Page 194
Summary……Page 195
Counting derrees of freedom……Page 196
Finding the formula via the symmetry principie……Page 197
Interpreting the formula geometrically……Page 198
Introduction to Riemann´s Mapping Theorem……Page 200
Exercises……Page 201
A puzzling phenomenon……Page 209
Introduction……Page 211
The jacobian matrix……Page 212
The amplitwist concept……Page 213
The real derivative re-examined……Page 214
The complex derivative……Page 215
Analytic functions……Page 217
A brief summary……Page 218
Some simple examples……Page 219
Introduction……Page 220
Conformality throughout a region……Page 221
Conformality and the Riemann sphere……Page 223
Degrees of crushing……Page 224
Breakdown of conformality……Page 225
Branch points……Page 226
Introduction……Page 227
The geometry of linear transformations……Page 228
The Cauchy-Riemann equations……Page 229
Exercises……Page 231
The cartesian form……Page 236
The polar form……Page 237
An intimation of rigidity……Page 239
Visual differentiation of log(z)……Page 242
Composition……Page 243
Inverse functions……Page 244
Addition and multiplication……Page 245
Polynomials……Page 246
Power series……Page 247
Rational functions……Page 248
Visual differentiation of the power function……Page 249
Visual differentiation of exp(z)……Page 251
Geometric solution of E´=E……Page 252
Introduction……Page 254
Analytic transformation of curvature……Page 255
Complex curvature……Page 258
Two kinds of elliptical orbit……Page 261
Changing the first into the second……Page 263
The geometry of force……Page 264
An explanation……Page 265
The Kasner-Arnold´s theorem……Page 266
Introduction……Page 267
Rigidity……Page 269
Uniqueness……Page 270
Preservation of indentities……Page 271
Analytic continuation via reflections……Page 272
Exercises……Page 278
The parallel axiom……Page 287
Some facts from non-euclidean geometry……Page 289
Geometry on a curved surface……Page 291
Gaussian curvature……Page 293
Surfaces of constant curvature……Page 295
The connection with Möbius transformations……Page 297
The angular excess of a spherical triangle……Page 298
Motions of the sphere……Page 299
A conformal map of the sphere……Page 303
Spatial rotations as Möbius transformations……Page 306
Spatial Rotations and quaternions……Page 310
The tractix and the pseudosphere……Page 313
The constant curvature of the pseudosphere……Page 315
A conformal map of the pseudosphere……Page 316
Beltrami´s hiperbolic plane……Page 318
Hiperbolic lines and reflections……Page 321
The Bolyai-Lobachevsky formula……Page 325
The three types of direct motion……Page 326
Decomposition into two reflections……Page 331
The angular excess of a hiperbolic triangle……Page 333
The Poincare disc……Page 336
Motions of the Poincaré disc……Page 339
The hemisphere model and hyperbolic space……Page 342
Exercises……Page 348
Definition……Page 358
What does “inside” mean?……Page 359
Finding winding numbers quickly……Page 360
The result……Page 361
Loops as mappings of the circle*……Page 362
The explanation*……Page 363
Polynomials and the argument principie……Page 364
Counting preimages algebraically……Page 366
Counting preimages geometrically……Page 367
Topological characteristics of analyticity……Page 369
A topological argument principie……Page 370
Two examples……Page 372
The result……Page 373
Brouwer´s fixed point theorem*……Page 374
Maximum-modulus theorem……Page 375
Schwarz´s lemma……Page 377
Liouville´s theorem……Page 379
Pick´s result……Page 380
Rational functions……Page 383
Poles and essential singularities……Page 385
The explanation*……Page 387
Exercises……Page 389
Introduction……Page 397
The Riemann sum……Page 398
The trapezoidal rule……Page 399
Geometric estimation of errors……Page 400
Complex Riemann sums……Page 403
A useful inequality……Page 406
Rules of integration……Page 407
A circular arc……Page 408
General loops……Page 410
Winding number……Page 411
Introduction……Page 412
Area interpretation……Page 413
Integration along a circular arc……Page 415
General contours and the deformation theorem……Page 417
A further extension of the theorem……Page 419
Residues……Page 420
The exponential mapping……Page 421
Introduction……Page 422
An example……Page 423
The fundamental theorem……Page 424
The integral as antiderivate……Page 426
Logaritm as integral……Page 428
Parametric evaluation……Page 429
Some preliminaries……Page 430
The explanation……Page 432
The result……Page 434
The explanation……Page 435
A simpler explanation……Page 437
The general formula of contour integration……Page 438
Exercises……Page 440
First explanation……Page 447
General Cauchy formula……Page 449
Infinity differentiability……Page 451
Taylor series……Page 452
Laurent series centred at a pole……Page 454
A formula for calculating residues……Page 455
Application to real integrals……Page 456
Calculating residues using taylor series……Page 458
Application to summation of series……Page 459
Laurent´s theorem……Page 462
Exercises……Page 466
Complex functions as vector fields……Page 470
Physical vector fields……Page 471
Flows and force fields……Page 473
Sources and sinks……Page 474
The index of a singular point……Page 476
The index according to Poincaré……Page 479
The index theorem……Page 480
Formulation of the Poincaré-Hopf theorem……Page 482
Defining the index on a surface……Page 484
An explanation fo the Poincaré-Hopf theorem……Page 485
Exercises……Page 488
Flux……Page 492
Work……Page 494
Local flux and local work……Page 496
Divergence and crul in geometric form*……Page 498
Divergence-free and crul-free vector fields……Page 499
The Pólya vector field……Page 501
Cauchy´s theorem……Page 503
Example: Area as flux……Page 504
Example: Winding number as flux……Page 505
Local behaviour of vector fields*……Page 506
Cauchy´s formula……Page 508
Positive powers……Page 509
Negative powers and multipoles……Page 510
Multipoles at infinity……Page 512
Laurent´s series as a multipole expansion……Page 513
The stream function……Page 514
The gradient field……Page 517
The potential function……Page 518
The complex potential function……Page 520
Examples……Page 523
Exercises……Page 525
Dual flows……Page 528
Harmonic duals……Page 531
Conformal invariance of harmonicity……Page 533
Conformal invariance of the Laplacian……Page 535
The meaning fo the Laplacian……Page 536
A powerful computational tool……Page 537
The curvature of harmonic equipotentials……Page 540
Further complex curvature calculations……Page 543
Further geometry of the complex curvature……Page 545
Introduction……Page 547
An example……Page 548
The metoth of images……Page 552
Mapping one flow onto another……Page 558
Introduction……Page 560
Exterior mappings and flows round obstacles……Page 561
Interior mappings and dipoles……Page 564
Interior mappings, vortices, and sources……Page 566
An example: automorphisms of the disc……Page 569
Green´s function……Page 570
Introduction……Page 574
Schwarz´s interpretation……Page 576
Dirichlet´s problem for the disc……Page 578
The interpretations of Neumann and Böcher……Page 580
Green general formula……Page 585
Exercises……Page 590
References……Page 593
Index……Page 599
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