目錄
Preface
Symbols and Terms
Preliminaries
1.1 Preview
A It Takes Two Harmonic Functions
B Heat Flow
C A Geometric Rule
D Electrostatics
E Fluid Flow
F One Model Many Applications
Exercises
1.2 Sets, Functions, and Visualization
A Terminology and Notation for Sets
B Terminology and Notation for Functions
C Functions from R to R
D Functions from R2 to R
E Functions from R2 to R2
Exercises
1.3 Structures on R2, and Linear Maps from R2 to R2
A The Real Line and the Plane
B Polar Coordinates in the Plane
C When Is a Mapping M : R2 → R2 Linear?
D Visualizing Nonsingular Linear Mappings
E The Determinant of a Two-by-Two Matrix
F Pure Magnifications, Rotations, and
Conjugation
G Conformal Linear Mappings
Exercises
1.4 Open Sets, Open Mappings, Connected Sets
A Distance, Interior, Boundary, Openness
B Continuity in Terms of Open Sets
C Open Mappings
D Connected Sets
Exercises
1.5 A Review of Some Calculus
A Integration Theory for Real- Valued Functions
B Improper Integrals, Principal Values
C Partial Derivatives
D Divergence and Curl
Exercises
1.6 Harmonic Functions
A The Geometry of Laplace's Equation
B The Geometry of the Cauchy-Riemann
Equations
C The Mean Value Property
D Changing Variables in a Dirichtet or Neumann
Problem
Exercises
2 Basic Tools
2.1 The Complex Plane
A The Definition of a Field
B Complex Multiplication
C Powers and Roots
D Conjugation
E Quotients of Complex Numbers
F When Is a Mapping L : C → C Linear?
G Complex Equations for Lines and Circles
H The Reciprocal Map, and Reflection in the
Unit Circle
I Reflections in Lines and Circles
Exercises
2.2 Visualizing Powers, Exponential, Logarithm, and Sine
A Powers ofz
B Exponential and Logarithms
C Sin z
D The Cosine and Sine, and the Hyperbolic
Cosine and Sine
Exercises
2.3 Differentiability
A Differentiability at a Point
B Differentiability in the Complex Sense:
Holomorphy
C Finding Derivatives
D Picturing the Local Behavior of Holomorphic
Mappings
Exercises
2.4 Sequences, Compactness, Convergence
A Sequences of Complex Numbers
B The Limit Superior of a Sequence of Reals
C Implications of Compactness
D Sequences of Functions
Exercises
2.5 Integrals Over Curves, Paths, and Contours
A Integrals of Complex-Valued Functions
B Curves
C Paths
D Pathwise Connected Sets
E Independence of Path and Morera's Theorem
F Goursat's Lemma
G The Winding Number
H Green's Theorem
I Irrotational and Incompressible Fluid Flow
J Contours
Exercises
2.6 Power Series
A Infinite Series
B The Geometric Series
C An Improved Root Test
D Power Series and the Cauchy-Hadamard
Theorem
E Uniqueness of the Power Series Representation
F Integrals That Give Rise to Power Series
Exercises
3 The Cauchy Theory
3.1 Fundamental Properties of Holomorphic Functions
A Integral and Series Representations
B Eight Ways to Say "'Holomorphic"
C Determinism
D Liouville's Theorem
E The Fundamental Theorem of Algebra
F Subuniform Convergence Preserves
Holomorphy
Exercises
3.2 Cauchy's Theorem
A Cerny's 1976 Proof
B Simply Connected Sets
C Subuniform Boundedness, Subuniform
Convergence
3.3 lsolated Singularities
A The Laurent Series Representation on an Annulus
B Behavior Near an Isolated Singularity in the Plane
C Examples: Classifying Singularities, Finding Residues
D Behavior Near a Singularity at Infinity
E A Digression: Picard'sGreat Theorem
Exercises
3.4 The Residue Theorem and the Argument Principle
A Meromorphic Functions and the Extended
Plane
B The Residue Theorem
C Multiplicity and Valence
D Valence.for a Rational Function
E The Argument Principle: Integrals That Count
Exercises
3.5 Mapping Properties
Exercises
3.6 The Riemann Sphere
Exercises
4 The Residue Calculus
4.1 Integrals of Trigonometric Functions Over a Compact lnterval
Exercises
4.2 Estimating Complex Integrals
Exercises
4.3 Integrals of Rational Functions Over the Line
Exercises
4.4 Integrals Involving the Exponential
A Integrals Giving Fourier Transforms
Exercises
4.5 Integrals Involving a Logarithm
Exercises
4.6 Integration on a Riemann Surface
A Mellin Transforms
Exercises
4. 7 The Complex Inversion Formula for the Laplace Transform
Exercises
5 Boundary Value Problems
5.1 Examples
A Easy Problems
B The Conformal Mapping Method
Exercises
5.2 The Mobius Maps
Exercises
5.3 Electric Fields
A A Point Charge in 3-Space
B Uniform Charge on One or More Long Wires
C Examples with Bounded Potentials
Exercises
5.4 Steady Flow of a Perfect Fluid
Exercises
5.5 Using the Poisson Integral to Obtain Solutions
A The Poisson Integral on a Disk
B Solutions on the Disk by the Poisson Integral
C Geometry of the Poisson Integral
D Harmonic Functions and the Mean Value Property
E The Neumann Problem on a Disk
F The Poisson Integral on a Half-Plane, and on Other Domains
Exercises
5.6 When Is the Solution Unique?
Exercises
5.7 The Schwarz Reflection Principle
5.8 Schwarz-Christoffel Formulas
A Triangles
B Rectangles and Other Polygons
C Generalized Polygons
Exercises
6 Lagniappe
6.1 Dixon's 1971ProofofCauchy's Theorem
6.2 Runge's Theorem
Exercises
6.3 The Riemann Mapping Theorem
Exercises
6.4 The Osgood-Taylor-Carath~odory Theorem
References
Index
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