Visual Complex Analysis

Cover......Page 1
Visual Complex Analysis - Tristan Needham (OUP, 1997)......Page 2
Preface......Page 15
Acknowledgements......Page 20
Contents......Page 6
Historical Skentch......Page 22
Bombelli's 'Wild Thought'......Page 24
Some terminology and notation......Page 27
Practice......Page 28
Equivalence of Symbolic and geometric arithmetic......Page 29
Introduction......Page 31
Moving particle argument......Page 32
Power series argument......Page 33
Introduction......Page 35
Trigonometry......Page 36
Geometry......Page 37
Calculus......Page 41
Algebra......Page 43
Vectorial operations......Page 48
Geometry through the eyes of Felix Klein......Page 51
Classifying motions......Page 55
Three reflections theorem......Page 58
Similarities and Complex arithmetic......Page 60
Spatial complex numers?......Page 64
Excercises......Page 66
Introduction......Page 76
Positive Integer Powers......Page 78
Cubics revisited *......Page 80
Cassinian Curves *......Page 81
The mystery of real power series......Page 85
The disc of convergence......Page 88
Approximating a power series with a polynomial......Page 91
Uniqueness......Page 92
Manipulating power series......Page 93
Finding the radius of convergence......Page 95
Fourier series*......Page 98
Power series approach......Page 100
The geometry of the mapping......Page 101
Another approach......Page 102
Definitions and identities......Page 105
Relation to hyperbolic functions......Page 107
The geometry of the mapping......Page 109
Example: Fractional powers......Page 111
Single-valued branches of a multifunction......Page 113
Relevance to power series......Page 116
An example with two branch points......Page 117
Inverse of the exponential function......Page 119
The logarithmic power series......Page 121
General powers......Page 122
The centroid......Page 123
Averaging over regular polygons......Page 126
Averaging over circles......Page 129
Exercises......Page 132
Connection with Einstein's theory of relativity*......Page 143
Preliminary definitions and facts......Page 145
Preservation of circles......Page 147
Construction using orthogonal circles......Page 150
Preservation of angles......Page 151
Inversion in a sphere......Page 154
A problem on touching circles......Page 157
Quadrilaterals with orthogonal diagonals......Page 158
Ptolemy's theorem......Page 159
The point at infinity......Page 160
Stereografic projection......Page 161
Transferring complex functions to the sphere......Page 164
Behaviour of functions at infinity......Page 165
Stereographic formulae......Page 167
Preservation of circles, angles and symmetry......Page 169
Non-uniqueness of the coefficients......Page 170
The group property......Page 171
Fixed points......Page 172
Fixed points at infinity......Page 173
The cross-ratio......Page 175
Evidence of a link with linear algebra......Page 177
The explanation: Homogeneous coordinates......Page 178
Eigenvectors and eigenvalues......Page 180
Rotations of the sphere......Page 182
The main idea......Page 183
Elliptic, hiperbolic, and loxodromic types......Page 185
Local geometric inerpretation of the multipler......Page 187
Parabolic transformations......Page 189
Computing the multipler......Page 190
Eingenvalue interpretation of the multipler......Page 191
Elliptic case......Page 193
Hyperbolic case......Page 194
Parabolic case......Page 195
Summary......Page 196
Counting derrees of freedom......Page 197
Finding the formula via the symmetry principie......Page 198
Interpreting the formula geometrically......Page 199
Introduction to Riemann's Mapping Theorem......Page 201
Exercises......Page 202
A puzzling phenomenon......Page 210
Introduction......Page 212
The jacobian matrix......Page 213
The amplitwist concept......Page 214
The real derivative re-examined......Page 215
The complex derivative......Page 216
Analytic functions......Page 218
A brief summary......Page 219
Some simple examples......Page 220
Introduction......Page 221
Conformality throughout a region......Page 222
Conformality and the Riemann sphere......Page 224
Degrees of crushing......Page 225
Breakdown of conformality......Page 226
Branch points......Page 227
Introduction......Page 228
The geometry of linear transformations......Page 229
The Cauchy-Riemann equations......Page 230
Exercises......Page 232
The cartesian form......Page 237
The polar form......Page 238
An intimation of rigidity......Page 240
Visual differentiation of log(z)......Page 243
Composition......Page 244
Inverse functions......Page 245
Addition and multiplication......Page 246
Polynomials......Page 247
Power series......Page 248
Rational functions......Page 249
Visual differentiation of the power function......Page 250
Visual differentiation of exp(z)......Page 252
Geometric solution of E'=E......Page 253
Introduction......Page 255
Analytic transformation of curvature......Page 256
Complex curvature......Page 259
Two kinds of elliptical orbit......Page 262
Changing the first into the second......Page 264
The geometry of force......Page 265
An explanation......Page 266
The Kasner-Arnold's theorem......Page 267
Introduction......Page 268
Rigidity......Page 270
Uniqueness......Page 271
Preservation of indentities......Page 272
Analytic continuation via reflections......Page 273
Exercises......Page 279
The parallel axiom......Page 288
Some facts from non-euclidean geometry......Page 290
Geometry on a curved surface......Page 292
Gaussian curvature......Page 294
Surfaces of constant curvature......Page 296
The connection with Moebius transformations......Page 298
The angular excess of a spherical triangle......Page 299
Motions of the sphere......Page 300
A conformal map of the sphere......Page 304
Spatial rotations as Moebius transformations......Page 307
Spatial Rotations and quaternions......Page 311
The tractix and the pseudosphere......Page 314
The constant curvature of the pseudosphere......Page 316
A conformal map of the pseudosphere......Page 317
Beltrami's hiperbolic plane......Page 319
Hiperbolic lines and reflections......Page 322
The Bolyai-Lobachevsky formula......Page 326
The three types of direct motion......Page 327
Decomposition into two reflections......Page 332
The angular excess of a hiperbolic triangle......Page 334
The Poincare disc......Page 337
Motions of the Poincare disc......Page 340
The hemisphere model and hyperbolic space......Page 343
Exercises......Page 349
Definition......Page 359
What does 'inside' mean?......Page 360
Finding winding numbers quickly......Page 361
The result......Page 362
Loops as mappings of the circle*......Page 363
The explanation*......Page 364
Polynomials and the argument principie......Page 365
Counting preimages algebraically......Page 367
Counting preimages geometrically......Page 368
Topological characteristics of analyticity......Page 370
A topological argument principie......Page 371
Two examples......Page 373
The result......Page 374
Brouwer's fixed point theorem*......Page 375
Maximum-modulus theorem......Page 376
Schwarz's lemma......Page 378
Liouville's theorem......Page 380
Pick's result......Page 381
Rational functions......Page 384
Poles and essential singularities......Page 386
The explanation*......Page 388
Exercises......Page 390
Introduction......Page 398
The Riemann sum......Page 399
The trapezoidal rule......Page 400
Geometric estimation of errors......Page 401
Complex Riemann sums......Page 404
A useful inequality......Page 407
Rules of integration......Page 408
A circular arc......Page 409
General loops......Page 411
Winding number......Page 412
Introduction......Page 413
Area interpretation......Page 414
Integration along a circular arc......Page 416
General contours and the deformation theorem......Page 418
A further extension of the theorem......Page 420
Residues......Page 421
The exponential mapping......Page 422
Introduction......Page 423
An example......Page 424
The fundamental theorem......Page 425
The integral as antiderivate......Page 427
Logaritm as integral......Page 429
Parametric evaluation......Page 430
Some preliminaries......Page 431
The explanation......Page 433
The result......Page 435
The explanation......Page 436
A simpler explanation......Page 438
The general formula of contour integration......Page 439
Exercises......Page 441
First explanation......Page 448
General Cauchy formula......Page 450
Infinity differentiability......Page 452
Taylor series......Page 453
Laurent series centred at a pole......Page 455
A formula for calculating residues......Page 456
Application to real integrals......Page 457
Calculating residues using taylor series......Page 459
Application to summation of series......Page 460
Laurent's theorem......Page 463
Exercises......Page 467
Complex functions as vector fields......Page 471
Physical vector fields......Page 472
Flows and force fields......Page 474
Sources and sinks......Page 475
The index of a singular point......Page 477
The index according to Poincare......Page 480
The index theorem......Page 481
Formulation of the Poincare-Hopf theorem......Page 483
Defining the index on a surface......Page 485
An explanation fo the Poincare-Hopf theorem......Page 486
Exercises......Page 489
Flux......Page 493
Work......Page 495
Local flux and local work......Page 497
Divergence and crul in geometric form*......Page 499
Divergence-free and crul-free vector fields......Page 500
The Polya vector field......Page 502
Cauchy's theorem......Page 504
Example: Area as flux......Page 505
Example: Winding number as flux......Page 506
Local behaviour of vector fields*......Page 507
Cauchy's formula......Page 509
Positive powers......Page 510
Negative powers and multipoles......Page 511
Multipoles at infinity......Page 513
Laurent's series as a multipole expansion......Page 514
The stream function......Page 515
The gradient field......Page 518
The potential function......Page 519
The complex potential function......Page 521
Examples......Page 524
Exercises......Page 526
Dual flows......Page 529
Harmonic duals......Page 532
Conformal invariance of harmonicity......Page 534
Conformal invariance of the Laplacian......Page 536
The meaning fo the Laplacian......Page 537
A powerful computational tool......Page 538
The curvature of harmonic equipotentials......Page 541
Further complex curvature calculations......Page 544
Further geometry of the complex curvature......Page 546
Introduction......Page 548
An example......Page 549
The metoth of images......Page 553
Mapping one flow onto another......Page 559
Introduction......Page 561
Exterior mappings and flows round obstacles......Page 562
Interior mappings and dipoles......Page 565
Interior mappings, vortices, and sources......Page 567
An example: automorphisms of the disc......Page 570
Green's function......Page 571
Introduction......Page 575
Schwarz's interpretation......Page 577
Dirichlet's problem for the disc......Page 579
The interpretations of Neumann and Boecher......Page 581
Green general formula......Page 586
Exercises......Page 591
References......Page 594
Index......Page 600