Visual complex analysis = 复分析 : 可视化方法 /

1 geometry and complex arithmetic　1.

i introduction　1

1　historical sketch　1

2　bombelli's “wild thought”　3

3　some terminology and notation　6

4　practice　7

5　symbolic and geometric arithmetic　8

ii euler's formula　10

1　introduction　10

2　moving particle argument　10

3　power series argument　12

4　sine and cosine in terms of euler's formula　14

iii some applications　14

1　introduction　14

2　trigonometry　14

3　geometry　16

4　calculus　20

5　algebra　22

6　vectorial operations　27

iv transformations and euclidean geometry*　30

.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...