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
