By H. Begehr

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Because both functions coincide on r they are analytic continuations from one another forming together an entire analytic function with pole at infinity of order at most x. From the general LIOUVILLE theorem this function is seen to be a polynomical P. of degree at most x. Thus, z E D+ , 0+(z) = P"(z)e"+(z), (z) = z-"P"(z)e" (z), z E D- . ii. x

It is enough to assume it, v E C2(D; IR) n C' (D;IR) for the second Green formula. In the sequal we return to the two-dimensional case although the next theorem holds in higher dimensions, too and again use complex variables. Theorem 13. (GREEN). Let D C t be a bounded domain with (piecewise) smooth boundary and having a GREEN function g(z, zo) Then for any harmonic function u which on aD is still continuous u(z) = -2x fr u(()aanz)dst t (z E D) aD This is the GREEN representation formula for harmonic functions.

Find an analytic function in d' \r satisfying on 9(S) G(C)S r. 't'his problem is called RIEMANN problem with shift or delay, too. For a solution see [Gakh66], p. 121. It is solved by reducing the problem to the RIEMANN problem. There is still a lot of new research done on RIEMANN and related boundary value problems. Next we will consider the RIEMANN problem for a multiply connected domain. Let r,,, 0 < µ < m, be m + 1 mutually disjoint smooth simply closed curves such that ro positively oriented surrounds the other rµ(1 < p < m) being negatively oriented.

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

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