More Explorations in Complex Functions is something of a sequel to GTM 287, Explorations in Complex Functions. Both texts introduce a variety of topics, from core material in the mainstream of complex analysis to tools that are widely used in other areas of mathematics and applications, but there is minimal overlap between the two books. The intended readership is the same, namely graduate students and researchers in complex analysis, independent readers, seminar attendees, or instructors for a second course in complex analysis. Instructors will appreciate the many options for constructing a second course that builds on a standard first course in complex analysis. Exercises complement the results throughout. There is more material in this present text than one could expect to cover in a year’s course in complex analysis. A mapping of dependence relations among chapters enables instructors and independent readers a choice of pathway to reading the text. Chapters 2, 4, 5, 7, and 8 contain the function theory background for some stochastic equations of current interest, such as SLE.
The text begins with two introductory chapters to be used as a resource. Chapters 3 and 4 are stand-alone introductions to complex dynamics and to univalent function theory, including deBrange’s theorem, respectively. Chapters 5—7 may be treated as a unit that leads from harmonic functions to covering surfaces to the uniformization theorem and Fuchsian groups. Chapter 8 is a stand-alone treatment of quasiconformal mapping that paves the way for Chapter 9, an introduction to Teichmüller theory. The final chapters, 10–14, are largely stand-alone introductions to topics of both theoretical and applied interest: the Bergman kernel, theta functions and Jacobi inversion, Padé approximants and continued fractions, the Riemann—Hilbert problem and integral equations, and Darboux’s method for computing asymptotics.
Author(s): Richard Beals, Roderick S.C. Wong
Series: Graduate Texts in Mathematics 298
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 403
City: Cham
Tags: Complex Dynamics, Univalent Functions, (Sub-)Harmonic Functions, Riemann Surfaces, Uniformization Theorem, Quasiconformal Mappings, Teichmüller Theory, Bergman Kernel, Theta Functions, Continued Fractions, Riemann-Hilbert Problems
Preface
Contents
1 Basics
1.1 Introduction; notation
1.2 The Cauchy–Riemann equations and Cauchy's integral theorem
1.3 The Cauchy integral formula and applications
1.4 Change of contour, isolated singularities, residues
1.5 The logarithm and powers
1.6 Infinite products
1.7 Reflection principles
1.8 Analytic continuation
1.9 Harmonic functions
Remarks and further reading
2 Further preliminaries
2.1 Linear fractional transformations
2.2 Geometries
2.3 Normal families
2.4 Conformal equivalence and the Riemann mapping theorem
2.5 The triply-punctured sphere, Montel, and Picard
2.6 Jordan domains and Carathéodory's extension theorem
2.7 Hilbert spaces
2.8 Lp spaces and measure
2.9 Convolution, approximation, and weak solutions
2.10 The gamma function
Remarks and further reading
3 Complex dynamics
3.1 Fatou sets and Julia sets; some examples
3.2 Julia sets: invariance, density, and self-similarity
3.3 Fixed points and periodic points
3.4 Attracting, super-attracting, and repelling fixed points
3.5 Neutral fixed points
3.6 Parabolic fixed points
3.7 Perspectives: classification and the Mandelbrot set
Exercises
Remarks and further reading
4 Univalent functions and de Branges's theorem
4.1 Bieberbach's theorem and some consequences
4.2 The Bieberbach conjecture: history and strategy
4.3 The Carathéodory convergence theorem
4.4 Slit mappings and Loewner's equation
4.5 The Robertson and Milin conjectures
4.6 Preparation for the proof of de Branges's theorem
4.7 Proof of de Branges's Theorem
Exercises
Remarks and further reading
5 Harmonic and subharmonic functions; the Dirichlet problem
5.1 Harmonic functions and the Poisson integral formula
5.2 Harnack's principle; removable singularities
5.3 Subharmonic functions and Perron's principle
5.4 Regular points and the solution of the Dirichlet problem
5.5 The L2 approach to the Dirichlet problem
Exercises
Remarks and further reading
6 General Riemann surfaces
6.1 Abstract Riemann surfaces
6.2 The universal cover
6.3 Automorphism groups and cover transformations
Exercises
Remarks and further reading
7 The uniformization theorem
7.1 Green's functions and harmonic measure
7.2 Uniformization: the hyperbolic case
7.3 An analogue of the Green's function
7.4 Proof of the uniformization theorem, completed
Exercises
Remarks and further reading
8 Quasiconformal mapping
8.1 Quadrilaterals
8.2 Quasiconformal mappings
8.3 Regular quasiconformal maps
8.4 Ring domains
8.5 Extremal ring domains
8.6 Distortion properties and Hölder continuity
8.7 Quasisymmetry and quasi-isometry
8.8 Complex dilatation; the Beltrami equation
8.9 The Calderón–Zygmund inequality
Exercises
Remarks and further reading
9 Introduction to Teichmüller theory
9.1 Coverings, quotients, and moduli of compact Riemann surfaces
9.2 Homeomorphisms of Riemann surfaces
9.3 Homeomorphisms of compact Riemann surfaces
9.4 The Teichmüller space of a Riemann surface
9.5 The universal Teichmüller space
9.6 The Bers embedding
9.7 Further developments
9.8 Higher Teichmüller theory
Exercises
Remarks and further reading
10 The Bergman kernel
10.1 The reproducing kernel
10.2 Orthonormal bases
10.3 Conformal mapping, I
10.4 Conformal invariance and the Bergman metric
10.5 Conformal mapping, II
10.6 The kernel function and partial differential equations
Exercises
Remarks and further reading
11 Theta functions
11.1 Hyperelliptic curves
11.2 Cycles and differentials
11.3 Theta functions and Abel's theorem
11.4 Jacobi inversion
Exercises
Remarks and further reading
12 Padé approximants and continued fractions
12.1 Padé approximants and Taylor series
12.2 Padé approximation and continued fractions
12.3 Another view of Padé approximants and continued fractions
12.4 The Stieltjes transform, Padé approximants, and orthogonal polynomials
12.5 Characterization of Stieltjes transforms
12.6 Stieltjes functions and Padé approximants
12.7 Generalized Shanks Transformation
12.8 Examples
12.9 Continued fraction expansions of ex
Exercises
Remarks and further reading
13 Riemann–Hilbert problems
13.1 The Sokhotski–Plemelj formula
13.2 Riemann–Hilbert Problems
13.3 The Radon Transform and the Fourier transform
13.4 Integral Equations with Cauchy Kernels
13.5 Integral Equations with Algebraic Kernels
13.6 Integral Equations with Logarithmic Kernels
13.7 Singular Integral Equations
13.8 The other Riemann–Hilbert problem
Exercises
Remarks and further reading
14 Asymptotics and Darboux's method
14.1 Algebraic singularities
14.2 Logarithmic singularities
14.3 Two coalescing singularities
14.4 Asymptotic nature of the expansion (14.3.24)
14.5 Heisenberg polynomials
Exercises
Remarks and further reading
Appendix References
Index
