By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)
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Example text
We thus also take into consideration the possibility that the "region" Aw may consist solely of the arcs 1 2 "'w· § 1. The Principle of Harmonic Measure 39 is harmonic and single-valued inc:. Regarding the boundary points z* there are four cases to consider. 1. z* lies within G, and therefore belongs to the set Oi,. Here w{z*, For various questions, the logarithm of the quotient of the noneuclidean and euclidean lengths of an arc element plays an important role. log l~ir, then u is uniquely defined in G and indeed independent of how the image point x = x(z) of z is chosen under the auxiliary mapping G00 -+ K. \Ve have If one sets u(z, G) = da = e"(z,G) [dz[ = eu(x,K) )dx[ for the value of the hyperbolic length da of dz and dx, which is invariant relative to the mapping mentioned. ::~! 2 . 3), it is clear that L1 zu(z ' G) = L1 zu(x , K) = L1 x u(x ' K) I dx ( .. We thus also take into consideration the possibility that the "region" Aw may consist solely of the arcs 1 2 "'w· § 1. The Principle of Harmonic Measure 39 is harmonic and single-valued inc:. Regarding the boundary points z* there are four cases to consider. 1. z* lies within G, and therefore belongs to the set Oi,. Here w{z*,