Visual complex functions: an introduction with phase portraits

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Автор: Elias Wegert

Название: Visual complex functions: an introduction with phase portraits

Язык: English

Издательство: Basel; New York: Birkhäuser/Springer

Год: 2012

Формат: pdf

Размер: 20,6 mb

Страниц: 360

This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains.

Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes.

In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions.

Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

Preface ix

1 Getting Acquainted 1

2 Complex Functions 13

2.1 Complex Numbers.................................................. 14

2.2 Functions and Mappings........................................... 23

2.3 Arithmetic and Geometry.......................................... 25

2.4 The Analytic Landscape........................................... 27

2.5 Color Representations............................................ 29

2.6 Convergence and Continuity....................................... 42

2.7 Some Plane Geometry.............................................. 45

3 Analytic Functions 59

3.1 Polynomials and Rational Functions............................... 60

3.2 Power Series..................................................... 72

3.3 Introduction to Analytic Functions............................... 94

3.4 Analytic Functions in Planar Domains............................. 99

3.5 Analytic Functions on the Sphere.................................112

3.6 Analytic Continuation............................................117

4 Complex Calculus 133

4.1 Complex Differentiation..........................................134

4.2 Complex Integration..............................................150

4.3 Cauchy Integral Formula..........................................169

4.4 Laurent Series and Singularities.................................175

4.5 Residues.........................................................183

4.6 Conjugate Harmonic Functions.....................................191

5 Construction Principles 203

5.1 Function Sequences...............................................203

5.2 Normal Families .................................................206

5.3 Function Series.................................................210

5.4 Infinite Products...............................................219

5.5 Cauchy Integrals................................................227

5.6 Integrals with Parameters.......................................241

6 Conformal Mappings 253

6.1 Mappings of Planar Domains......................................254

6.2 Special Conformal Mappings .....................................259

6.3 Mobius Transformations..........................................271

6.4 The Riemann Mapping Theorem.....................................278

6.5 Boundary Correspondence.........................................283

6.6 The Reflection Principle........................................289

6.7 Elliptic Integrals..............................................295

6.8 The Schwarz-Christoffel Formula.................................302

7 Riemann Surfaces 311

7.1 Global Analytic Functions.......................................312

7.2 Lifting Techniques..............................................314

7.3 Typical Examples................................................317

7.4 Analytic Functions and Branch Points............................322

7.5 Abstract Riemann Surfaces.......................................330

7.6 Practical Excursion.............................................337

Epilogue 345

Bibliography 347

Index 352