By Christian Pommerenke
We learn the boundary behaviour of a conformal map of the unit disk onto an arbitrary easily hooked up aircraft area. A crucial objective of the speculation is to acquire a one-to-one correspondence among analytic homes of the functionality and geometrie homes of the area. within the classical purposes of conformal mapping, the area is bounded by way of a piecewise soft curve. in lots of fresh purposes although, the area has a really undesirable boundary. it might probably have nowhere a tangent as is the case for Julia units. Then the conformal map has many unforeseen homes, for example just about all the boundary is mapped onto virtually not anything and vice versa. The ebook is intended for 2 teams of clients. (1) Graduate scholars and others who, at a variety of degrees, are looking to find out about conformal mapping. so much sections include workouts to check the comprehend ing. they generally tend to be relatively uncomplicated and just a couple of include new fabric. Pre specifications are common genuine and intricate analyis together with the fundamental proof approximately conformal mapping (e.g. AhI66a). (2) Non-experts who are looking to get an idea of a specific point of confor mal mapping to be able to locate whatever invaluable for his or her paintings. such a lot chapters for that reason commence with an outline that states a few key effects heading off tech nicalities. The booklet isn't really intended as an exhaustive survey of conformal mapping. a number of vital points needed to be passed over, e.g. numerical tools (see e.g.
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8 for z E 171" eit + z -'t-cp(t) dt 27r -71" e' - z + -i g(z) == log[(z - 1)'" f(z)] = a1 (18) ]]J) , li and thus 171" eitcp(t) I 1171" Icp(t)1 = ;: -71" (eit _ z)2 dt :S;: -71" leit _ Zl2 dt. 6) that, for 0 2c Ig'(r)l:S 7r l 1- r 0 < r < 1 - 8, (1 - r)'" 2c ( )2 dt+ 1- r 7r 2t'" jO -2-dt+M1 171" t'" 2'dt 7r tot 1-r = (2c + 2c7r ) (1- r)"'-l + M 2 (c). 7r Since 0 1- Q < Q < 1 it follows by integration that Ig(l) - g(r)1 :S cM3 (1 - r)'" + M 2 (c)(1 - r). Hence g(r) = g(l)+o((l-r)"') as r -> 1-, and (17), (18) show that Img(l) = 7rQ.
R >O If ! is continuous in [J) it follows that C(f, () is compact and connected and thus either a continuum or a single point. Let now E c [J) and ( E 1l' n E. The cluster set CE(f, () of! (Zn) --+ w as n --+ 00 . We shall be interested in the cases that E is a Stolz angle Ll at ( or that E is a curve r in [J) ending at (. , () are connected compact sets if ! is continuous in [J). See the books Nos60 and CoL066 for the theory of cluster sets. 34 Chapter 2. Continuity and Prime Ends The next theorem (CarI3b, Linl5) relates the geometrically defined prime ends with the analytically defined cluster sets.
If we define