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Analytic

By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)

ISBN-10: 364285592X

ISBN-13: 9783642855924

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Sample text

The values w that w(z) assumes in G, lie in a region Gw bounded by finitely many I ordan arcs 2. ,. ,. 1) where G! ,. Proof. Let z be a point where w(z) lies outside of Aw, and suppose G~ is that well-determined subregion of G. 2) R. NEVANLINNA [12]. 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.

If it were now true that

This function is also harmonic, since the latter property is invariant with respect to any conformal transformation of the variable. 2. In order to solve the boundary value problem we have posed in the more general case where the frontier of G is composed of several different Jordan curves Fv ... , Fp, we call upon the conformal mapping investigated in I, § 4, of the universal covering surface of the multiply connected region G onto the unit disk K. This mapping is produced by a linear polymorphic function x = x(z), and indeed in such a way that each curve is associated with one single boundary arc in the case p = 2, and in the case p > 2, with an infinite sequence of disjoint, open arcs on the circumference [x[ = 1 (cf.

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