By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)
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Extra info for Analytic Functions
These zero points are branch points of the level lines w · J.. \Ve first of all prove that these branch points cannot accumulate toward the boundary F. For this purpose we take a boundary point t; lying on the outermost boundary curve rl. Ux > 0. The function
There are thus only two possibilities: either the level lines terminate at the end points of IX, or they are closed, in which case they must enclose one or more boundary (like any closed subarc of a level line). arcs \Ve now consider the analytic function rv rv rp(z, eX, G) = w(z, eX, G) + i w(z, IX, G), where w stands for the conjugate of w; w is uniquely determined up to an additive constant. On any closed path in a multiply connected region (p > 1), w experiences an increase that depends linearly on the p - 1 periodicity modules associated with the inner boundary curves 1'2 , ••• , Fp.
Finally, the integral (:U) is a harmonic function in G. )LJ,w 1 as n-+ =, where LJ,w(v = 1, ... , n) is the harmonic measure of a system sn of subarcs on the boundary on which the points ~v are chosen arbitrarily; the arcs inS, are selected so that their diameters (the maximum distance between two points on one and the same arc) vanish as n --c>- =· This approximation is a harmonic function of zinG, and since the com·ergence is uniform in every interior subregion of Gl, the harmonicity of the r.