.

0 (hodnocen0 x )

BK

1st ed.

Oxford : Clarendon Press, 2005

xxiii,592 s. : il.

ISBN 0-19-853446-9 (brož.)

Obsahuje ilustrace, předmluvu, rejstřík

Bibliografie na s. 573-578

Analýza komplexní - učebnice vysokošk.

000119818

Contents // 1 Geometry and Complex Arithmetic 1 // I Introduction 1 // 1 Historical Sketch 1 // 2 Bombellľs "Wild Thoughf 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 // 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 // xvi Contents // 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 Möbius Transformations and Inversion 122 // I Introduction 122 // 1 Definition of Möbius Transformations 122 // 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 // Contents xvii // 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 Möbius 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 Möbius 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---ЕхогŃ�ізоэ---tet4 Differentiation: The Amplitwist Concept 189 // I Introduction 189 // II A Puzzling Phenomenon 189 // III Local Description of Mappings in the Plane 191 // 1 Introduction 191 // 2 The Jacobian Matrix 192 // 3 The Amplitwist Concept 193 // xviii Contents // 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 // VIII 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 216 // 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 Functions 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 // VIII Geometric Solution of E’ = E 232 // IX An Application of Higher Derivatives: Curvature* 234 // 1 Introduction 234 // Contents xix // 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-ArnoÄľd Theorem 246 // XI Analytic Continuation*

247 // 1 Introduction 247 // 2 Rigidity 249 // 3 Uniqueness 250 // 4 Preservation of Identities 251 // 5 Analytic Continuation via Reflections 252 // XII 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 Möbius 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 Möbius 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 // XX Contents // 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 Rouché’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 // VIII 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 // Contents xxi // 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 // VIII 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 // XI The General Cauchy Theorem 414 // 1 The Result 414 // 2 The Explanation 415 // 3 A Simpler Explanation 417 // XII The General Formula of Contour Integration 418 // XIII 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 // xxii Contents // 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 P ’.ya 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 // Contents xxiii // 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 Invariance 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 Bocher 560 // 5 Green’s General Formula 565 // VIII Exercises 570 // References 573 // Index // 579