Download e-book for kindle: Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B.

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

ISBN-10: 3642855903

ISBN-13: 9783642855900

ISBN-10: 364285592X

ISBN-13: 9783642855924

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Example text

4. The example considered in 1. 1) becomes an equality, even for multivalent mappings. Let us assume for a moment that this is the case at an interior point z of G,. ) - w(z, IX=, G=) is definitely nonnegative, it must vanish identically by the minimum principle. , one can always achieve a situation where w tends to an interior point (of continuity} of x w; afterwan;ls one can again let the r egion tend to the original Aw. 40 III. ). 3) 'When G. and G! 3) ? The results of II, § 5, yield a complete answer to this question.

1 To achieve this it is sufficient for the boundary function u[() to be continum1sly if is analytic. differentiable with respect to the arc length of r. r 30 II. 1) points to a relation between the function h and the harmonic measure w(z, cx) of an arc cx = cx(() bounded by an arbitrarily chosen fixed boundary point ( 0 and by a movable boundary point (. 2), that is, if one sets u(C) = 1 for all the points on ex and u(C) = 0 for all the other boundary points, w(z, ex) = 21 J dh(C, z) = 1! ioit 1 2 (h(C 1 , z) ~ h(C0 , zJ).

On a, w assumes the value 1, and on the complementary arc {3, the value 0. In the following we shall investigate the behavior of the harmonic measure when the given region G is subjected to a conformal mapping. If the region G + F is mapped one-to-one onto the region G' + F' so that the mapping is conformal on the interior of G and continuous on the boundary F =IX+ p, the arcs IX, Pgoing to the arcs ex', {3', the harmonic measure w(z, ex, G) is transformed into a function of the image point z' of z.

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


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