By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)
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Nevertheless, if we confine our interest to a subregion g >A > 0, where A is chosen large enough for this subregion GA to be simply connected, then the above exponential function yields a one-to-one, conformal mapping of GA onto the disk JtJ < e-A, from which it is clear that in the region GA the h-lines exhibit the simple behavior described above. Thus, even for a multiply connected region it is true that the h-lines running out from the pole are related to values in the interval (0, 2n) in such a way that lines which correspond to values h1 and h 2 bound an angle of magnitude h2 - h1 .
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). 1! The harmonic measure w(z, ex) is thus equal to 2~ times the increase experienced by the harmonic function ~h(C, z) conjugate to the Green's function g(C, z) (with respect to C), when Cmoves along the arc cx in the negative direction.
4. , zero) when z-+ oo along the real axis. Then for a given arbitrarily small 0 < e (< 1), as soon as z passes a certain point z = t0 on the positive real axis, f w(z) I < s. ) < l. Now it is immediately clear that the harmonic measure w is equal to 1/n times the magnitude of the angle formed by the line through z and ! 0 and that part of the real axis lying to the right of t0 as seen from z (nw = n - arg (z- t0 )), and the region A< w < 1 is thus an angle whose vertex lies at z = t0 and which is bounded by the positive real axis and the half ray arg (z - t0 ) = n(l -A).
Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)