Complex Analytic Methods for Partial Diff. Eqns. - An Intro. by H. Begehr

By H. Begehr

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Let z(V) be the inverse mapping of yo = ep(z) mapping D schlicht onto D. f(z) ='p(z) (z E D) . Theorem 8. Let w(z, zo) be the RIEMANN mapping function from the domain D onto the unit disc D for fixed zo E D, satisfying w(zo, zo) = 0, (zo, zo) > 0 . Then 9(z,zo):=-loglw(z,zo)I, ZED, has the following properties. i. 9(z,zo) is harmonic in z E D\{zo}, ii. log Iz - zol + g(z, zo) is harmonic in the neighborhood of zo, iii. lim g(z, zo) = 0. zBD Remark. This result can be reverted. If g(z, zo) is the GREEN function of a simply connected domain D, see Definition 3, and h(z,zo) its harmonic conjugate, see p.

Now L iv'. r N(z(s), zo)a(s)ds = 0, zo E D, 0 is demanded, see [Hawe72], p. 113, [Wend79], p. 5. The NEUMANN function is sometimes called the second GREEN function, g(z, zo) the first GREEN function and Gf (z, zo) := g(z, z0), G"(z, zo) := N(z, zo) used. Remark. A regular curve is an open or closed continuously differentiable curve r:= {z :z=z(t),o

Moreover, this result is true for any point of D (of V \D) replacing z = 0. Let w(z) be a conformal mapping from D onto itself mapping zo onto 0. Then P(z) := ico + >{ckwk(z) - Ckw-k(z)}, co E JR, ck E 0 , I < k < n , k=1 is a solution of the homogeneous DIRICHLET problem with a pole at zo, and zo cannot be a zero of P if P is not identically zero. This even holds when w is the conformal mapping from some domain onto D. Corollary 1. The general solution to the DIRICHLET problem for analytic functions in the unit disc D is w(z) -z S 2fii co E 1R .

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