复分析--可视化方法(英文影印版)

.1　geometry through the eyes of felix klein　30
2　classifying motions　34
3　three reflections theorem　37
4　similarities and complex arithmetic　39
5　spatial complex numbers?　43
v exercises　45
2　complex functions as transformations　55
i introduction　55
ii polynomials　57
1　positive integer powers　57
2　cubics revisited*　59
3　cassinian curves*　60
iii power series　64
1　the mystery of real power series　64
2　the disc of convergence　67
3　approximating a power series with a polynomial　70
4　uniqueness　71
5　manipulating power series　72
6　finding the radius of convergence　74
7　fourier series*　77
iv the exponential function　79
1　power series approach　79
2　the geometry of the mapping　80
3　another approach　81
v cosine and sine　84
1　definitions and identities　84
2　relation to hyperbolic functions　86
3　the geometry of the mapping　88
vi multifunctions　90
1　example: fractional powers　90
2　single-valued branches of a multifunction　92
3　relevance to power series　95
4　an example with two branch points　96
vii　the logarithm function　98
1　inverse of the exponential function　98
2　the logarithmic power series　100
3　general powers　101
viii　averaging over circles*　102
1　the centroid　102
2　averaging over regular polygons　105
3　averaging over circles　108
ix exercises　111
3 mobius transformations and inversion　122
i　introduction　122
1　definition of m6bius transformations122
2　connection with einstein's theory of relativity*　122
3　decomposition into simple transformations　 123
ii inversion　124
1　preliminary definitions and facts　124
2　preservation of circles　126
3　construction using orthogonal circles　128
4　preservation of angles　130
5　preservation of symmetry　133
6　inversion in a sphere　133
iii three illustrative applications of inversion　136
1　a problem on touching circles　136
2　quadrilaterals with orthogonal diagonals　137
3　ptolemy's theorem　138
iv the riemann sphere　139
1　the point at infinity　139
2　stereographic projection　140
3　transferring complex functions to the sphere　143
4　behaviour of functions at infinity　144
5　stereographic formulae*　146
v mobius transformations: basic results　148
1　preservation of circles, angles, and symmetry　148
2　non-uniqueness of the coefficients　149
3　the group property　150
4　fixed points　151
5　fixed points at infinity　152
6　the cross-ratio　154
vi mobius transformations as matrices*　156
1　evidence of a link with linear algebra　156
2　the explanation: homogeneous coordinates　157
3　eigenvectors and eigenvalues*　158
4　rotations of the sphere*　161
vii　visualization and classification*　162
1　the main idea　162
2　elliptic, hyperbolic, and loxodromic types　164
3　local geometric interpretation of the multiplier　166
4　parabolic transformations　168
5　computing the multiplier*　169
6　eigenvalue interpretation of the multiplier*　170
viii　decomposition into 2 or 4 reflections*　172
1　introduction　172
2　elliptic case　172
3　hyperbolic case　173
4　parabolic case　174
5　summary　175
ix automorphisms of the unit disc*　176
1　counting degrees of freedom　176
2　finding the formula via the symmetry principle　177
3　interpreting the formula geometrically*　178
4　introduction to riemann's mapping theorem　180
x exercises　181
4　differentiation: the amplitwist concept　189
i introduction　189
ii a puzzling phenomenon　189
iii local description of mappings in the plane　191
1　introduction　1 91
2　the jacobian matrix　192
3　the amplitwist concept　193
iv the complex derivative as amplitwist　194
1　the real derivative re-examined　194
2　the complex derivative　195
3　analytic functions　197
4　a brief summary　198
v some simple examples　199
vi conformal = analytic　200
1　introduction　200
2　conformality throughout a region　201
3　conformality and the riemann sphere　203
vii　critical points　204
1　degrees of crushing　204
2　breakdown of conformality　205
3　branch points　206
vlll　the cauchy-riemann equations　207
1　introduction　207
2　the geometry of linear transformations　208
3　the cauchy-riemann equations　209
ix exercises　211
5　further geometry of differentiation　216
i cauchy-riemann revealed　216
1　introduction　215
2　the cartesian form　216
3　the polar form　217
ii an intimation of rigidity　219
iii visual differentiation of log(z)　222
iv rules of differentiation　223
1　composition　223
2　inverse functions　224
3　addition and multiplication　225
v polynomials, power series, and rational func-tions　226
1　polynomials　226
2　power series　227
3　rational functions　228
vi visual differentiation of the power function　229
vii　visual differentiation of exp(z)　231
vlll　geometric solution of e'= e232
ix an application of higher derivatives: curva-ture*　234
1　introduction　234
2　analytic transformation of curvature　235
3　complex curvature　238
x celestial mechanics*　241
1　central force fields　241
2　two kinds of elliptical orbit　241
3　changing the first into the second　243
4　the geometry of force　244
5　an explanation　245
6　the kasner-arnol'd theorem　246
xl analytic continuation*　247
1　introduction　247
2 rigidity　249
3　uniqueness　250
4　preservation of identities　251
5　analytic continuation via reflections　252
xll　exercises　258
6　non-euclidean geometry*　267..
i introduction　267
1　the parallel axiom　267
2　some facts from non-euclidean geometry　269
3　geometry on a curved surface　270
4　intrinsic versus extrinsic geometry　273
5　gaussian curvature　273
6　surfaces of constant curvature　275
7　the connection with m6bius transformations　277
ii spherical geometry　278
1　the angular excess of a spherical triangle　278
2　motions of the sphere　279
3　a conformal map of the sphere　283
4　spatial rotations as m6bius transformations　286
5　spatial rotations and quaternions　290
iii hyperbolic geometry　293
1　the tractrix and the pseudosphere　293
2　the constant curvature of the pseudosphere*　295
3　a conformal map of the pseudosphere　296
4　beltrami's hyperbolic plane　298
5　hyperbolic lines and reflections　301
6　the bolyai-lobachevsky formula*　305
7　the three types of direct motion　306
8　decomposition into two reflections　311
9　the angular excess of a hyperbolic triangle　313
10　the poincaré disc　315
11　motions of the poincaré disc　319
12　the hemisphere model and hyperbolic space　322
iv exercises　328
7　winding numbers and topology　338
i　winding number　338
1　the definition　338
2　what does “inside” mean?　339
3　finding winding numbers quickly　340
ii hopf's degree theorem　341
1　the result　341
2　loops as mappings of the circle*　342
3　the explanation*　343
iii polynomials and the argument principle　344
iv a topological argument principle*　346
1　counting preimages algebraically　346
2　counting preimages geometrically　347
3　topological characteristics of analyticity　349
4　a topological argument principle　350
5　two examples　352
v rouche's theorem　353
1　the result　353
2　the fundamental theorem of algebra　354
3　brouwer's fixed point theorem*　354
vi maxima and minima　355
1　maximum-modulus theorem　355
2　related results　357
vii　the schwarz-pick lemma*　357
1　schwarz's lemma　357
2　liouville's theorem　359
3　pick's result　360
vlll　the generalized argument principle　363
1　rational functions　363
2　poles and essential singularities　365
3　the explanation*　367
ix exercises　369
8　complex integration: cauchy's theorem　377
i introduction　377
ii the real integral　378
1　the riemann sum　378
2　the trapezoidal rule　379
3　geometric estimation of errors　380
iii the complex integral　383
1　complex riemann sums　383
2　a visual technique　386
3　a useful inequality　386
4　rules of integration　387
iv complex inversion　388
1　a circular arc　388
2　general loops　390
3　winding number　391
v conjugation　392
1　introduction　392
2　area interpretation　393
3　general loops　395
vi power functions　395
1　integration along a circular arc　395
2　complex inversion as a limiting case*　397
3　general contours and the deformation theorem　397
4　a further extension of the theorem　399
5　residues　400
vii　the exponential mapping　401
vlll　the fundamental theorem　402
1　introduction　402
2　an example　403
3　the fundamental theorem　404
4　the integral as antiderivative　406
5　logarithm as integral　408
ix parametric evaluation　409
x cauchy's theorem　410
1　some preliminaries　410
2　the explanation　412
xl the general cauchy theorem　414
1　the result　414
2　the explanation　415
3　a simpler explanation　417
xll　the general formula of contour integration 418
xlii　exercises　420
9　cauchy's formula and its applications　427
i cauchy's formula　427
1　introduction　427
2　first explanation　427
3　gauss' mean value theorem　429
4　general cauchy formula　429
ii infinite differentiability and taylor series　431
1　infinite differentiability　431
2　taylor series　432
iii calculus of residues　434
1　laurent series centred at a pole　434
2　a formula for calculating residues　435
3　application to real integrals　436
4　calculating residues using taylor series　438
5　application to summation of series　439
iv annular laurent series　442
1　an example　442
2　laurent's theorem　442
v exercises　446
10　vector fields: physics and topology　450
i vector fields　450
1　complex functions as vector fields　450
2　physical vector fields　451
3　flows and force fields　453
4　sources and sinks　454
ii winding numbers and vector fields*　456
1　the index of a singular point　456
2　the index according to poincaré　459
3　the index theorem　460
iii flows on closed surfaces*　462
1　formulation of the poincaré-hopf theorem　462
2　defining the index on a surface　464
3　an explanation of the poincaré-hopf theorem　465
iv exercises　468
11　vector fields and complex integration　472
i flux and work　472
1　flux　472
2　work　474
3　local flux and local work　476
4　divergence and curl in geometric form*　478
5　divergence-free and curl-free vector fields　479
ii complex integration in terms of vector fields 481
1　the p61ya vector field　481
2　cauchy's theorem　483
3　example: area as flux　484
4　example: winding number as flux　485
5　local behaviour of vector fields*　486
6　cauchy's formula　488
7　positive powers　489
8　negative powers and multipoles　490
9　multipoles at infinity　492
10　laurent's series as a multipole expansion　493
iii the complex potential　494
1　introduction　494
2　the stream function　494
3　the gradient field　497
4　the potential function　498
5　the complex potential　500
6　examples　503
iv exercises　505
12　flows and harmonic functions　508
i harmonic duals　508
1　dual flows　508
2　harmonic duals　511
ii conformal i nvariance　513
1　conformal invariance of harmonicity　513
2　conformal invariance of the laplacian　515
3　the meaning of the laplacian　516
iii a powerful computational tool　517
iv the complex curvature revisited*　520
1　some geometry of harmonic equipotentials　520
2　the curvature of harmonic equipotentials　520
3　further complex curvature calculations　523
4　further geometry of the complex curvature　525
v flow around an obstacle　527
1　introduction　527
2　an example　527
3　the method of images　532
4　mapping one flow onto another　538
vi the physics of riemann's mapping theorem 540
1　introduction　540
2　exterior mappings and flows round obstacles　541
3　interior mappings and dipoles　544
4　interior mappings, vortices, and sources　546
5　an example: automorphisms of the disc　549
6　green's function　550
vii dirichlet's problem　554
1　introduction　554
2　schwarz's interpretation　556
3　dirichlet's problem for the disc　558
4　the interpretations of neumann and bscher　560
5　green's general formula　565
vlll　exercises　570
references　573
index　579...