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单复变函数(第二卷 英文版)2011年版

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  • 大小:13.4 MB
  • 语言:中文版
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资源简介
单复变函数(第二卷 英文版)
出版时间:2011年版
内容简介
  This is the sequel to my book Functions of One Complex Variable I, and probably a good opportunity to express my appreciation to the mathemat-ical community for its reception of that work. In retrospect, writing that book was a crazy venture.As a graduate student I had had one of the worst learning experiences of my career when I took complex analysis; a truly bad teacher. As a non-tenured assistant professor, the department allowed me to teach the graduate course in complex analysis. They thought I knew the material; I wanted to learn it. I adopted a standard text and shortly after beginning to prepare my lectures I became dissatisfied. All the books in print had virtues; but I was educated as a modern analyst, not a classical one, and they failed to satisfy me.
目录
preface
13 return to basics
 1 regions and curves
 2 derivatives and other recollections
 3 harmonic conjugates and primitives
 4 analytic arcs and the reflection principle
 5 boundary values for bounded analytic functions
14 conformal equivalence for simply connected regions
 1 elementary properties and examples
 2 crosscuts
 3 prime ends
 4 impressions of a prime end
 5 boundary values of riemann maps.
 6 the area theorem.
 7 disk mappings: the class $
15 conformal equivalence for finitely connected regions
 1 analysis on a finitely connected region.
 2 conformal equivalence with an analytic jordan region
 3 boundary values for a conformed equivalence between finitelyconnected jordan regions  
 4 convergence of univalent functions
 5 conformed equivalence with a circularly slit annulus
 6 conformal equivalence with a circularly slit disk.
 7 conformal equivalence with a circular region
16 analytic covering maps
 1 results for abstract covering spaces
 2 analytic covering spaces
 3 the modular function
 4 applications of the modular function.
 5 the existence of the universal analytic covering map
17 de branges's proof of the bieberbach conjecture
 1 subordination
 2 loewner chains
 3 loewner's differential equation
 4 the milin conjecture
 5 some special functions
 6 the proof of de branges's theorem
18 some fundamental concepts from analysis
 1 bergman spaces of analytic and harmonic functions
 2 partitions of unity.
 3 convolution in euclidean space
 4 distributions
 5 the cauchy transform
 6 an application: rational approximation
 7 fourier series and cesaro sums
19 harmonic functions redux
 1 harmonic functions on the disk
 2 fatou's theorem
 3 semicontinuous functions
 4 subharmonic functions.
 5 the logarithmic potential
 6 an application: approximation by harmonic functions
 7 the dirichlet problem
 8 harmonic majorants
 9 the green function
 10 regular points for the dirichlet problem.
 11 the dirichlet principle and sobolev spaces
20 hardy spaces on the disk
 1 definitions and elementary properties
 2 the nevanlinna class
 3 factorization of functions in the nevanlinna class
 4 the disk algebra
 5 the invariant subspaces of hp
 6 szegs's theorem
21 potential theory in the plane
 1 harmonic measure
 2 the sweep of a measure
 3 the robin constant
 4 the green potential
 5 polar sets
 6 more on regular points
 7 logarithmic capacity: part 1
 8 some applications and examples of logarithmic capacity
 9 removable singularities for functions in the bergman space
 10 logarithmic capacity: part 2
 11 the transfinite diameter and logarithmic capacity
 12 the refinement of a subharmonic function
 13 the fine topology.
 14 wiener's criterion for regular points
contents
references
list of symbols
index
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