Complex Analysis by Stewart, Ian -- Read -- Imperial Library of Trantor

Index

Cover Half-title page Title page Imprints page Contents Preface to the Second Edition Preface to the First Edition 0 The Origins of Complex Analysis, and Its Challenge to Intuition

0.1 The Origins of Complex Numbers 0.2 The Origins of Complex Analysis 0.3 The Puzzle 0.4 Is Mathematics Discovered or Invented? 0.5 Overview of the Book

1 Algebra of the Complex Plane

1.1 Construction of the Complex Numbers 1.2 The Notation 1.3 A Geometric Interpretation 1.4 Real and Imaginary Parts 1.5 The Modulus 1.6 The Complex Conjugate 1.7 Polar Coordinates 1.8 The Complex Numbers Cannot be Ordered 1.9 Exercises

2 Topology of the Complex Plane

2.1 Open and Closed Sets 2.2 Limits of Functions 2.3 Continuity 2.4 Paths

2.4.1 Standard Paths 2.4.2 Visualising Paths 2.4.3 The Image of a Path

2.5 Change of Parameter

2.5.1 Preserving Direction

2.6 Subpaths and Sums of Paths 2.7 The Paving Lemma 2.8 Connectedness 2.9 Space-filling Curves 2.10 Exercises

3 Power Series

3.1 Sequences 3.2 Series 3.3 Power Series 3.4 Manipulating Power Series 3.5 Products of Series 3.6 Exercises

4 Differentiation

4.1 Basic Results 4.2 The Cauchy–Riemann Equations 4.3 Connected Sets and Differentiability 4.4 Hybrid Functions 4.5 Power Series 4.6 A Glimpse Into the Future

4.6.1 Real Functions Differentiable Only Finitely Many Times 4.6.2 Bad Behaviour of Real Taylor Series 4.6.3 The Blancmange function 4.6.4 Complex Analysis is Better Behaved

4.7 Exercises

5 The Exponential Function

5.1 The Exponential Function 5.2 Real Exponentials and Logarithms 5.3 Trigonometric Functions 5.4 An Analytic Definition of π 5.5 The Behaviour of Real Trigonometric Functions 5.6 Dynamic Explanation of Euler’s Formula 5.7 Complex Exponential and Trigonometric Functions are Periodic 5.8 Other Trigonometric Functions 5.9 Hyperbolic Functions 5.10 Exercises

6 Integration

6.1 The Real Case 6.2 Complex Integration Along a Smooth Path 6.3 The Length of a Path

6.3.1 Integral Formula for the Length of Smooth Paths and Contours

6.4 If You Took the Short Cut … 6.5 Further Properties of Lengths

6.5.1 Lengths of More General Paths

6.6 Regular Paths and Curves

6.6.1 Parametrisation by Arc Length

6.7 Regular and Singular Points 6.8 Contour Integration

6.8.1 Definition of Contour Integral

6.9 The Fundamental Theorem of Contour Integration 6.10 An Integral that Depends on the Path 6.11 The Gamma Function

6.11.1 Known Properties of the Gamma Function

6.12 The Estimation Lemma 6.13 Consequences of the Fundamental Theorem 6.14 Exercises

7 Angles, Logarithms, and the Winding Number

7.1 Radian Measure of Angles 7.2 The Argument of a Complex Number 7.3 The Complex Logarithm 7.4 The Winding Number 7.5 The Winding Number as an Integral 7.6 The Winding Number Round an Arbitrary Point 7.7 Components of the Complement of a Path 7.8 Computing the Winding Number by Eye 7.9 Exercises

8 Cauchy’s Theorem

8.1 The Cauchy Theorem for a Triangle 8.2 Existence of an Antiderivative in a Star Domain 8.3 An Example – the Logarithm 8.4 Local Existence of an Antiderivative 8.5 Cauchy’s Theorem 8.6 Applications of Cauchy’s Theorem

8.6.1 Cuts and Jordan Contours

8.7 Simply Connected Domains 8.8 Exercises

9 Homotopy Versions of Cauchy’s Theorem

9.1 Informal Description of Homotopy 9.2 Integration Along Arbitrary Paths 9.3 The Cauchy Theorem for a Boundary 9.4 Formal Definition of Homotopy 9.5 Fixed End Point Homotopy 9.6 Closed Path Homotopy 9.7 Converse to Cauchy’s Theorem 9.8 The Cauchy Theorems Compared 9.9 Exercises

10 Taylor Series

10.1 Cauchy Integral Formula 10.2 Taylor Series 10.3 Morera’s Theorem 10.4 Cauchy’s Estimate 10.5 Zeros 10.6 Extension Functions 10.7 Local Maxima and Minima 10.8 The Maximum Modulus Theorem 10.9 Exercises

11 Laurent Series

11.1 Series Involving Negative Powers 11.2 Isolated Singularities 11.3 Behaviour Near an Isolated Singularity 11.4 The Extended Complex Plane, or Riemann Sphere 11.5 Behaviour of a Differentiable Function at Infinity 11.6 Meromorphic Functions 11.7 Exercises

12 Residues

12.1 Cauchy’s Residue Theorem 12.2 Calculating Residues 12.3 Evaluation of Definite Integrals 12.4 Summation of Series 12.5 Counting Zeros 12.6 Exercises

13 Conformal Transformations

13.1 Measurement of Angles

13.1.1 Real Numbers Modulo 13.1.2 Geometry of 13.1.3 Operations on Angles 13.1.4 The Argument Modulo

13.2 Conformal Transformations 13.3 Critical Points 13.4 Möbius Maps

13.4.1 Möbius Maps Preserve Circles 13.4.2 Classification of Möbius Maps 13.4.3 Extension of Möbius Maps to the Riemann Sphere

13.5 Potential Theory

13.5.1 Laplace’s Equation 13.5.2 Design of Aerofoils

13.6 Exercises

14 Analytic Continuation

14.1 The Limitations of Power Series 14.2 Comparing Power Series 14.3 Analytic Continuation

14.3.1 Direct Analytic Continuation 14.3.2 Indirect Analytic Continuation 14.3.3 Complete Analytic Functions

14.4 Multiform Functions

14.4.1 The Logarithm as a Multiform Function 14.4.2 Singularities

14.5 Riemann Surfaces

14.5.1 Riemann Surface for the Logarithm 14.5.2 Riemann Surface for the Square Root 14.5.3 Constructing a General Riemann Surface by Gluing

14.6 Complex Powers 14.7 Conformal Maps Using Multiform Functions 14.8 Contour Integration of Multiform Functions 14.9 Exercises

15 Infinitesimals in Real and Complex Analysis

15.1 Infinitesimals 15.2 The Relationship Between Real and Complex Analysis

15.2.1 Critical Points

15.3 Interpreting Power Series Tending to Zero as Infinitesimals 15.4 Real Infinitesimals as Variable Points on a Number Line 15.5 Infinitesimals as Elements of an Ordered Field 15.6 Structure Theorem for any Ordered Extension Field of 15.7 Visualising Infinitesimals as Points on a Number Line 15.8 Complex Infinitesimals 15.9 Non-standard Analysis and Hyperreals 15.10 Outline of the Construction of Hyperreal Numbers 15.11 Hypercomplex Numbers 15.12 The Evolution of Meaning in Real and Complex Analysis

15.12.1 A Brief History 15.12.2 Non-standard Analysis in Mathematics Education 15.12.3 Human Visual Senses 15.12.4 Computer Graphics 15.12.5 Summary

15.13 Exercises

16 Homology Version of Cauchy’s Theorem

16.0.1 Outline of Chapter 16.0.2 Group-theoretic Interpretation 16.1 Chains 16.2 Cycles

16.2.1 Sums and Formal Sums of Paths

16.3 Boundaries 16.4 Homology 16.5 Proof of Cauchy’s Theorem, Homology Version

16.5.1 Grid of Rectangles 16.5.2 Proof of Theorem 16.2. 16.5.3 Rerouting Segments 16.5.4 Resumption of Proof of Theorem 16.2.

16.6 Cauchy’s Residue Theorem, Homology Version 16.7 Exercises

17 The Road Goes Ever On …

17.1 The Riemann Hypothesis 17.2 Modular Functions 17.3 Several Complex Variables 17.4 Complex Manifolds 17.5 Complex Dynamics 17.6 Epilogue

References Index

← Prev
Back
Next →

← Prev
Back
Next →