# Riemann Maps and World Maps

Chapter

## Abstract

Although the potential problem is linear, it is worth bearing in mind that
where, with the origin,
for appropriate complex numbers {

*infinite*-dimensional linear problems can be deeply difficult. Thus, consider finding Ø within the simply connected region Ω, where Δ Ø = 0 and Ø is given smoothly on the boundary, ∂Ω. Figure 1 significantly displays Ω to be processed of deep invaginations. A sharpening of ∂Ω there to a nondifferentiable “corner” would surely lead to a branch cut of Ø issuing out of it (see Figure 2). Thus, we should surmise that the continuation of Ø out of Ω will be something like Figure 3 with the “x”’s and “o”’s the poles and zeros the cuts have withered to upon smoothing ∂Ω. (Alternatively, the branch can persist, but detach from ∂Ω.) The dotted circle intersecting Ω up to the poles it passes through is of cardinal importance. Let us see why. Following (wrong headed) tradition, write Ø as a sum of harmonics. For ℜ^{2}(for which we immediately substitute*c*^{1}),$$\Delta {\psi _n} = 0 \Rightarrow {\psi _n} = (\operatorname{Re} ,\operatorname{Im} )({z^n})$$

(1)

*z =*0, within the interior of Ω,*n*is restricted to non-negative values. So we have$$\phi = \operatorname{Re} \sum\limits_0^\infty {{c_n}{z^n}}$$

(2)

*c*_{ n }}. As an approximation we sum to some*N -*1 and determine the 2*N*real parameters by calculating Ø on 2*N*suitably chosen points on ∂Ω. Presumably, end of problem.### Keywords

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© Springer Science+Business Media New York 1994