Visual Complex Analysis

Tristan NeedhamDepartment of MathematicsUniversity of San Francisco

CLARENDON PRESS • OXFORD

Contents

1 Geometry and Complex ArithmeticI Introduction

11

1 Historical Sketch 12 Bombelli's "Wild Thought" 33 Some Terminology and Notation 64 Practice 75 Symbolic and Geometric Arithmetic 8

II Euler's Formula 101 Introduction 102 Moving Particle Argument 103 Power Series Argument 124 Sine and Cosine in Terms of Euler's Formula 14

III Some Applications 141 Introduction 142 Trigonometry 143 Geometry 164 Calculus 205 Algebra 226 Vectorial Operations 27

IV Transformations and Euclidean Geometry* 301 Geometry Through the Eyes of Felix Klein 302 Classifying Motions 343 Three Reflections Theorem 374 Similarities and Complex Arithmetic 395 Spatial Complex Numbers? 43

V Exercises 45

2 Complex Functions as Transformations 55

1 Introduction 55II Polynomials 57

1 Positive Integer Powers 572 Cubics Revisited* 593 Cassinian Curves* 60

III Power Series 641 The Mystery of Real Power Series 642 The Disc of Convergence 67

xvi Contents

3 Approximating a Power Series with a Poiynomial4 Uniqueness5 Manipulating Power Series6 Finding the Radius of Convergence7 Fourier Series*

7071727477

IV The Exponential Function 791 Power Series Approach 792 The Geometry of the Mapping 803 Another Approach 81

V Cosine and Sine 841 Definitions and Identities 842 Relation to Hyperbolic Functions 863 The Geometry of the Mapping 88

VI Multifunctions 901 Example: Fractional Powers 902 Single-Valued Branches of a Multifunction 923 Relevance to Power Series 954 An Example with Two Branch Points 96

VII The Logarithm Function 981 Inverse of the Exponential Function 982 The Logarithmic Power Series 1003 General Powers 101

VIII Averaging over Circles* 1021 The Centroid 1022 Averaging over Regular Polygons 1053 Averaging over Circles 108

IX Exercises 111

3 Möbius Transformations and Inversion 122I Introduction 122

1 Definition of Möbius Transformations 1222 Connection with Einstein's Theory of Relativity* 1223 Decomposition into Simple Transformations 123

11 Inversion 1241 Preliminary Definitions and Facts 1242 Preservation of Circles 1263 Construction Using Orthogonal Circles 1284 Preservation of Angles 1305 Preservation of Symmetry 1336 Inversion in a Sphere 133

III Three Illustrative Applications of Inversion 1361 A Problem an Touching Circles 1362 Quadrilaterals with Orthogonal Diagonals 1373 Ptolemy's Theorem 138

Contents xvii

IV The Riemann Sphere 1391 The Point at Infinity 1392 Stereographic Projection 1403 Transferring Complex Functions to the Sphere 1434 Behaviour of Functions at Infinity 1445 Stereographic Formulae* 146

V Möbius Transformations: Basic Results 1481 Preservation of Circies, Angles, and Symmetry 1482 Non-Uniqueness of the Coefficients 1493 The Group Property 1504 Fixed Points 1515 Fixed Points at Infinity 1526 The Cross-Ratio 154

VI Möbius Transformations as Matrices* 1561 Evidence of a Link with Linear Algebra 1562 The Explanation: Homogeneous Coordinates 1573 Eigenvectors and Eigenvalues* 1584 Rotations of the Sphere* 161

VII Visualization and Classification* 1621 The Main Idea 1622 Elliptic, Hyperbolic, and Loxodromic types 1643 Local Geometric Interpretation of the Multiplier 1664 Parabolic Transformations 1685 Computing the Multiplier* 1696 Eigenvalue Interpretation of the Multiplier* 170

VIII Decomposition into 2 or 4 Reflections* 1721 Introduction 1722 Elliptic Case 1723 Hyperbolic Case 1734 Parabolic Case 1745 Summary 175

IX Automorphisms of the Unit Disc* 1761 Counting Degrees of Freedom 1762 Finding the Formula via the Symmetry Principle 1773 Interpreting the Formula Geometrically* 1784 Introduction to Riemann's Mapping Theorem 180

X Exercises 181

4 Differentiation: The Amplitwist Concept 189I Introduction 189II A Puzzling Phenomenon 189III Local Description of Mappings in the Plane 191

1 Introduction 1912 The Jacobian Matrix 1923 The Amplitwist Concept 193

xviii Contents

IV The Complex Derivative as Amplitwist1 The Real Derivative Re-examined2 The Complex Derivative3 Analytic Functions4 A Brief Summary

V Some Simple ExamplesVI Conformal = Analytic

1 Introduction2 Conformality Throughout a Region3 Conformality and the Riemann Sphere

VII Critical Points

194194195197198

199200200201203

2041 Degrees of Crushing 2042 Breakdown of Conformality 2053 Branch Points 206

VIII The Cauchy-Riemann Equations 2071 Introduction 2072 The Geometry of Linear Transformations 2083 The Cauchy-Riemann Equations 209

IX Exercises 211

5 Further Geometry of Differentiation 216I Cauchy-Riemann Revealed 216

1 Introduction 2162 The Cartesian Form 2163 The Polar Form 217

II An Intimation of Rigidity 219III Visual Differentiation of log(z) 222IV Rules of Differentiation 223

1 Composition 2232 Inverse Functions 2243 Addition and Multiplication 225

V Polynomials, Power Series, and Rational Func-tions 2261 Polynomials 2262 Power Series 2273 Rational Functions 228

VI Visual Differentiation of the Power Function 229VII Visual Differentiation of exp(z) 231VIII Geometric Solution of E' = E 232IX An Application of Higher Derivatives: Curva-

ture* 2341 Introduction 234

Contents xix

2 Analytic Transformation of Curvature 2353 Complex Curvature 238

X Celestial Mechanics* 2411 Central Force Fields 2412 Two Kinds of Elliptical Orbit 2413 Changing the First into the Second 2434 The Geometry of Force 2445 An Explanation 2456 The Kasner—Arnol'd Theorem 246

XI Analytic Continuation* 2471 Introduction 2472 Rigidity 2493 Uniqueness 2504 Preservation of Identities 2515 Analytic Continuation via Reflections 252

XII Exercises 258

6 Non-Euclidean Geometry* 267I Introduction 267

1 The Parallel Axiom 2672 Some Facts from Non-Euclidean Geometry 2693 Geometry an a Curved Surface 2704 Intrinsic versus Extrinsic Geometry 2735 Gaussian Curvature 2736 Surfaces of Constant Curvature 2757 The Connection with Möbius Transformations 277

II Spherical Geometry 2781 The Angular Excess of a Spherical Triangle 2782 Motions of the Sphere 2793 A Conformal Map of the Sphere 2834 Spatial Rotations as Möbius Transformations 2865 Spatial Rotations and Quaternions 290

III Hyperbolic Geometry 2931 The Tractrix and the Pseudosphere 2932 The Constant Curvature of the Pseudosphere* 2953 A Conformal Map of the Pseudosphere 2964 Beltrami's Hyperbolic Plane 2985 Hyperbolic Lines and Reflections 3016 The Bolyai-Lobachevsky Formula* 3057 The Three Types of Direct Motion 3068 Decomposition into Two Reflections 3119 The Angular Excess of a Hyperbolic Triangle 31310 The Poincar6 Disc 31511 Motions of the Poincar6 Disc 31912 The Hemisphere Model and Hyperbolic Space 322

IV Exercises 328

xx Contents

7 Winding Numbers and Topology 338

I Winding Number 3381 The Definition 3382 What does "inside" mean? 3393 Finding Winding Numbers Quickly 340

II Hopf's Degree Theorem 3411 The Result 3412 Loops as Mappings of the Circle* 3423 The Explanation* 343

III Polynomials and the Argument Principle 344IV A Topological Argument Principle* 346

1 Counting Preimages Algebraically 3462 Counting Preimages Geometrically 3473 Topological Characteristics of Analyticity 3494 A Topological Argument Principle 3505 Two Examples 352

V Rouchö's Theorem 3531 The Result 3532 The Fundamental Theorem of Algebra 3543 Brouwer's Fixed Point Theorem* 354

VI Maxima and Minima 3551 Maximum-Modulus Theorem 3552 Related Results 357

VII The Schwarz-Pick Lemma* 3571 Schwarz's Lemma 3572 Liouville's Theorem 3593 Pick's Result 360

VIII The Generalized Argument Principle 3631 Rational Functions 3632 Poles and Essential Singularities 3653 The Explanation* 367

IX Exercises 369

8 Complex Integration: Cauchy's Theorem 377

I Introduction 377II The Real Integral 378

1 The Riemann Sum 3782 The Trapezoidal Rule 3793 Geometric Estimation of Errors 380

III The Complex Integral 3831 Complex Riemann Sums 3832 A Visual Technique 3863 A Useful lnequality 386

Contents xxi

4 Rules of Integration 387IV Complex Inversion 388

1 A Circular Arc 3882 General Loops 3903 Winding Number 391

V Conjugation 3921 I ntroduction 3922 Area Interpretation 3933 General Loops 395

VI Power Functions 3951 Integration along a Circular Arc 3952 Complex Inversion as a Limiting Case* 3973 General Contours and the Deformation Theorem 3974 A Further Extension of the Theorem 3995 Residues 400

VII The Exponential Mapping 401VIII The Fundamental Theorem 402

1 I ntrod uction 4022 An Example 4033 The Fundamental Theorem 4044 The Integral as Antiderivative 4065 Logarithm as Integral 408

IX Parametric Evaluation 409X Cauchy's Theorem 410

1 Some Preliminaries 4102 The Explanation 412

XI The General Cauchy Theorem 4141 The Result 4142 The Explanation 4153 A Simpler Explanation 417

XII The General Formula of Contour Integration 418XIII Exercises 420

9 Cauchy's Formula and Its Applications 427

I Cauchy's Formula 4271 I ntroduction 4272 First Explanation 4273 Gauss' Mean Value Theorem 4294 General Cauchy Formula 429

II Infinite Differentiability and Taylor Series 4311 Infinite Differentiability 4312 Taylor Series 432

III Calculus of Residues 434

xxii Contents

1 Laurent Series Centred at a Pole 4342 A Formula for Calculating Residues 4353 Application to Real Integrals 4364 Calculating Residues using Taylor Series 4385 Application to Summation of Series 439

IV Annular Laurent Series 4421 An Example 4422 Laurent's Theorem 442

V Exercises 446

10 Vector Fields: Physics and Topology 450

1 Vector Fields 4501 Complex Functions as Vector Fields 4502 Physical Vector Fields 4513 Flows and Force Fields 4534 Sources and Sinks 454

11 Winding Numbers and Vector Fields* 4561 The Index of a Singular Point 4562 The Index According to Poincarö 4593 The Index Theorem 460

III Flows on Closed Surfaces* 4621 Formulation of the Poincarö-Hopf Theorem 4622 Defining the Index on a Surface 4643 An Explanation of the Poincarö-Hopf Theorem 465

IV Exercises 468

11 Vector Fields and Complex Integration 472I Flux and Work 472

1 Flux 4722 Work 4743 Local Flux and Local Work 4764 Divergence and Curl in Geometric Form* 4785 Divergence-Free and Curl-Free Vector Fields 479

II Complex Integrat;un in Terms of Vector Fields 4811 The Plya Vector Field 4812 Cauchy's Theorem 4833 Example: Area as Flux 4844 Example: Winding Number as Flux 4855 Local Behaviour of Vector Fields* 4866 Cauchy's Formula 4887 Positive Powers 4898 Negative Powers and Multipoles 4909 Multipoles at Infinity 49210 Laurent's Series as a Multipole Expansion 493

III The Complex Potential 494

Contents xxiii

1 Introduction 4942 The Stream Function 4943 The Gradient Field 4974 The Potential Function 4985 The Complex Potential 5006 Examples 503

IV Exercises 505

12 Flows and Harmonic Functions 508I Harmonic Duals 508

1 Dual Flows 5082 Harmonic Duals 511

II Conformal Invariance 5131 Conformal Invariance of Harmonicity 5132 Conformal Invariance of the Laplacian 5153 The Meaning of the Laplacian 516

III A Powerful Computational Tool 517IV The Complex Curvature Revisited* 520

1 Some Geometry of Harmonic Equipotentials 5202 The Curvature of Harmonic Equipotentials 5203 Further Complex Curvature Calculations 5234 Further Geometry of the Complex Curvature 525

V Flow Around an Obstacle 5271 Introduction 5272 An Example 5273 The Method of Images 5324 Mapping One Flow Onto Another 538

VI The Physics of Riemann's Mapping Theorem 5401 Introduction 5402 Exterior Mappings and Flows Round Obstacles 5413 Interior Mappings and Dipoles 5444 Interior Mappings, Vortices, and Sources 5465 An Example: Automorphisms of the Disc 5496 Green's Function 550

VII Dirichlet's Problem 5541 Introduction 5542 Schwarz's Interpretation 5563 Dirichlet's Problem for the Disc 5584 The lnterpretations of Neumann and Böcher 5605 Green's General Formula 565

VIII Exercises 570

References 573

Index 579