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By Reiner Kuhnau

Geometric functionality idea is a valuable a part of advanced research (one advanced variable). The guide of complicated research - Geometric functionality concept offers with this box and its many ramifications and family members to different parts of arithmetic and physics. the speculation of conformal and quasiconformal mappings performs a crucial function during this instruction manual, for instance a priori-estimates for those mappings which come up from fixing extremal difficulties, and positive tools are thought of. As a brand new box the speculation of circle packings which matches again to P. Koebe is integrated. The guide can be priceless for specialists in addition to for mathematicians operating in different components, in addition to for physicists and engineers.• a suite of self reliant survey articles within the box of GeometricFunction conception• lifestyles theorems and qualitative houses of conformal and quasiconformal mappings• A bibliography, together with many tricks to functions in electrostatics, warmth conduction, power flows (in the aircraft)

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3. Corners 52 . . . . . . . . . . . . . . . . . . . . . . . . . 4. Integral representations 4. General boundaries ......................................... 1. Distortion near the b o u n d a r y ...................................... 2. T h e angular derivative . . . . . . . . . . . . . . . . . . . . . 3. T h e b e h a v i o u r almost e v e r y w h e r e . . . . . . . . . . . . . . . . . .

1. T h e scope of this article ......................................... 2. T h r e e introductory examples 2. Continuity at the b o u n d a r y ...................................... 1. Jordan curves and locally c o n n e c t e d sets . . . . . . . . . . . . . . . . 2. Prime ends and cluster sets 43 ....................................... 3. Limits and injectivity . . . . . . . . . . . . . . . . . . . . . . 3. D o m a i n s with nice boundaries ........................................

2) for Itl - t2l ~< 6, where co increases and fo Jr co(x) dx < oo. 4. If B is a free Dini-smooth boundary arc then f ' has a continuous extension to 72 U A with f ' (z) 7~ O. , [123] for a proof of this theorem [59,121]. , [105, p. 48]. 4) Ch. 3). Then [121] the second derivative has a continuous extension to qI' U A. Let n = 1, 2 . . and 0 < ot < 1. We say that the smooth arc B is of class Cn'~ if the n-th derivative h (n) exists and satisfies the H61der condition h(n)(tl) -- h(n)(t2) - O(Itl - t2l~).

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