| |
| |
| Preface | |
| |
| |
| Symbols and Terms | |
| |
| |
| |
| Preliminaries | |
| |
| |
| |
| Preview | |
| |
| |
| |
| It Takes Two Harmonic Functions | |
| |
| |
| |
| Heat Flow | |
| |
| |
| |
| A Geometric Rule | |
| |
| |
| |
| Electrostatics | |
| |
| |
| |
| Fluid Flow | |
| |
| |
| |
| One Model, Many Applications | |
| |
| |
| Exercises | |
| |
| |
| |
| Sets, Functions, and Visualization | |
| |
| |
| |
| Terminology and Notation for Sets | |
| |
| |
| |
| Terminology and Notation for Functions | |
| |
| |
| |
| Functions from R to R | |
| |
| |
| |
| Functions from R[superscript 2] to R | |
| |
| |
| |
| Functions from R[superscript 2] to R[superscript 2] | |
| |
| |
| Exercises | |
| |
| |
| |
| Structures on R[superscript 2], and Linear Maps from R[superscript 2] to R[superscript 2] | |
| |
| |
| |
| The Real Line and the Plane | |
| |
| |
| |
| Polar Coordinates in the Plane | |
| |
| |
| |
| When Is a Mapping M: R[superscript 2] [right arrow] R[superscript 2] Linear? | |
| |
| |
| |
| Visualizing Nonsingular Linear Mappings | |
| |
| |
| |
| The Determinant of a Two-by-Two Matrix | |
| |
| |
| |
| Pure Magnifications, Rotations, and Conjugation | |
| |
| |
| |
| Conformal Linear Mappings | |
| |
| |
| Exercises | |
| |
| |
| |
| Open Sets, Open Mappings, Connected Sets | |
| |
| |
| |
| Distance, Interior, Boundary, Openness | |
| |
| |
| |
| Continuity in Terms of Open Sets | |
| |
| |
| |
| Open Mappings | |
| |
| |
| |
| Connected Sets | |
| |
| |
| Exercises | |
| |
| |
| |
| A Review of Some Calculus | |
| |
| |
| |
| Integration Theory for Real-Valued Functions | |
| |
| |
| |
| Improper Integrals, Principal Values | |
| |
| |
| |
| Partial Derivatives | |
| |
| |
| |
| Divergence and Curl | |
| |
| |
| Exercises | |
| |
| |
| |
| Harmonic Functions | |
| |
| |
| |
| The Geometry of Laplace's Equation | |
| |
| |
| |
| The Geometry of the Cauchy-Riemann Equations | |
| |
| |
| |
| The Mean Value Property | |
| |
| |
| |
| Changing Variables in a Dirichlet or Neumann Problem | |
| |
| |
| Exercises | |
| |
| |
| |
| Basic Tools | |
| |
| |
| |
| The Complex Plane | |
| |
| |
| |
| The Definition of a Field | |
| |
| |
| |
| Complex Multiplication | |
| |
| |
| |
| Powers and Roots | |
| |
| |
| |
| Conjugation | |
| |
| |
| |
| Quotients of Complex Numbers | |
| |
| |
| |
| When Is a Mapping L : C [right arrow] C Linear? | |
| |
| |
| |
| Complex Equations for Lines and Circles | |
| |
| |
| |
| The Reciprocal Map, and Reflection in the Unit Circle | |
| |
| |
| |
| Reflections in Lines and Circles | |
| |
| |
| Exercises | |
| |
| |
| |
| Visualizing Powers, Exponential, Logarithm, and Sine | |
| |
| |
| |
| Powers of z | |
| |
| |
| |
| Exponential and Logarithms | |
| |
| |
| |
| Sin z | |
| |
| |
| |
| The Cosine and Sine, and the Hyperbolic Cosine and Sine | |
| |
| |
| Exercises | |
| |
| |
| |
| Differentiability | |
| |
| |
| |
| Differentiability at a Point | |
| |
| |
| |
| Differentiability in the Complex Sense: Holomorphy | |
| |
| |
| |
| Finding Derivatives | |
| |
| |
| |
| Picturing the Local Behavior of Holomorphic Mappings | |
| |
| |
| Exercises | |
| |
| |
| |
| Sequences, Compactness, Convergence | |
| |
| |
| |
| Sequences of Complex Numbers | |
| |
| |
| |
| The Limit Superior of a Sequence of Reals | |
| |
| |
| |
| Implications of Compactness | |
| |
| |
| |
| Sequences of Functions | |
| |
| |
| Exercises | |
| |
| |
| |
| Integrals Over Curves, Paths, and Contours | |
| |
| |
| |
| Integrals of Complex-Valued Functions | |
| |
| |
| |
| Curves | |
| |
| |
| |
| Paths | |
| |
| |
| |
| Pathwise Connected Sets | |
| |
| |
| |
| Independence of Path and Morera's Theorem | |
| |
| |
| |
| Goursat's Lemma | |
| |
| |
| |
| The Winding Number | |
| |
| |
| |
| Green's Theorem | |
| |
| |
| |
| Irrotational and Incompressible Fluid Flow | |
| |
| |
| |
| Contours | |
| |
| |
| Exercises | |
| |
| |
| |
| Power Series | |
| |
| |
| |
| Infinite Series | |
| |
| |
| |
| The Geometric Series | |
| |
| |
| |
| An Improved Root Test | |
| |
| |
| |
| Power Series and the Cauchy-Hadamard Theorem | |
| |
| |
| |
| Uniqueness of the Power Series Representation | |
| |
| |
| |
| Integrals That Give Rise to Power Series | |
| |
| |
| Exercises | |
| |
| |
| |
| The Cauchy Theory | |
| |
| |
| |
| Fundamental Properties of Holomorphic Functions | |
| |
| |
| |
| Integral and Series Representations | |
| |
| |
| |
| Eight Ways to Say "Holomorphic" | |
| |
| |
| |
| Determinism | |
| |
| |
| |
| Liouville's Theorem | |
| |
| |
| |
| The Fundamental Theorem of Algebra | |
| |
| |
| |
| Subuniform Convergence Preserves Holomorphy | |
| |
| |
| Exercises | |
| |
| |
| |
| Cauchy's Theorem | |
| |
| |
| |
| Cerny's 1976 Proof | |
| |
| |
| |
| Simply Connected Sets | |
| |
| |
| |
| Subuniform Boundedness, Subuniform Convergence | |
| |
| |
| |
| Isolated Singularities | |
| |
| |
| |
| The Laurent Series Representation on an Annulus | |
| |
| |
| |
| Behavior Near an Isolated Singularity in the Plane | |
| |
| |
| |
| Examples: Classifying Singularities, Finding Residues | |
| |
| |
| |
| Behavior Near a Singularity at Infinity | |
| |
| |
| |
| A Digression: Picard's Great Theorem | |
| |
| |
| Exercises | |
| |
| |
| |
| The Residue Theorem and the Argument Principle | |
| |
| |
| |
| Meromorphic Functions and the Extended Plane | |
| |
| |
| |
| The Residue Theorem | |
| |
| |
| |
| Multiplicity and Valence | |
| |
| |
| |
| Valence for a Rational Function | |
| |
| |
| |
| The Argument Principle: Integrals That Count | |
| |
| |
| Exercises | |
| |
| |
| |
| Mapping Properties | |
| |
| |
| Exercises | |
| |
| |
| |
| The Riemann Sphere | |
| |
| |
| Exercises | |
| |
| |
| |
| The Residue Calculus | |
| |
| |
| |
| Integrals of Trigonometric Functions | |
| |
| |
| Exercises | |
| |
| |
| |
| Estimating Complex Integrals | |
| |
| |
| Exercises | |
| |
| |
| |
| Integrals of Rational Functions Over the Line | |
| |
| |
| Exercises | |
| |
| |
| |
| Integrals Involving the Exponential | |
| |
| |
| |
| Integrals Giving Fourier Transforms | |
| |
| |
| Exercises | |
| |
| |
| |
| Integrals Involving a Logarithm | |
| |
| |
| Exercises | |
| |
| |
| |
| Integration on a Riemann Surface | |
| |
| |
| |
| Mellin Transforms | |
| |
| |
| Exercises | |
| |
| |
| |
| The Inverse Laplace Transform | |
| |
| |
| Exercises | |
| |
| |
| |
| Boundary Value Problems | |
| |
| |
| |
| Examples | |
| |
| |
| |
| Easy Problems | |
| |
| |
| |
| The Conformal Mapping Method | |
| |
| |
| Exercises | |
| |
| |
| |
| The Mobius Maps | |
| |
| |
| Exercises | |
| |
| |
| |
| Electric Fields | |
| |
| |
| |
| A Point Charge in 3-Space | |
| |
| |
| |
| Uniform Charge on One or More Long Wires | |
| |
| |
| |
| Examples with Bounded Potentials | |
| |
| |
| Exercises | |
| |
| |
| |
| Steady Flow of a Perfect Fluid | |
| |
| |
| Exercises | |
| |
| |
| |
| Using the Poisson Integral to Obtain Solutions | |
| |
| |
| |
| The Poisson Integral on a Disk | |
| |
| |
| |
| Solutions on the Disk by the Poisson Integral | |
| |
| |
| |
| Geometry of the Poisson Integral | |
| |
| |
| |
| Harmonic Functions and the Mean Value Property | |
| |
| |
| |
| The Neumann Problem on a Disk | |
| |
| |
| |
| The Poisson Integral on a Half-Plane, and on Other Domains | |
| |
| |
| Exercises | |
| |
| |
| |
| When Is the Solution Unique? | |
| |
| |
| Exercises | |
| |
| |
| |
| The Schwarz Reflection Principle | |
| |
| |
| |
| Schwarz-Christoffel Formulas | |
| |
| |
| |
| Triangles | |
| |
| |
| |
| Rectangles and Other Polygons | |
| |
| |
| |
| Generalized Polygons | |
| |
| |
| Exercises | |
| |
| |
| |
| Lagniappe | |
| |
| |
| |
| Dixon's 1971 Proof of Cauchy's Theorem | |
| |
| |
| |
| Runge's Theorem | |
| |
| |
| Exercises | |
| |
| |
| |
| The Riemann Mapping Theorem | |
| |
| |
| Exercises | |
| |
| |
| |
| The Osgood-Taylor-Caratheodory Theorem | |
| |
| |
| References | |
| |
| |
| Index | |