More Explorations in Complex Functions

Description

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. 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