Riemann Maps and World Maps

  • Mitchell J. Feigenbaum
Part of the Applied Mathematical Sciences book series (AMS, volume 100)


Although the potential problem is linear, it is worth bearing in mind that 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 ℜ2 (for which we immediately substitute c1),
$$\Delta {\psi _n} = 0 \Rightarrow {\psi _n} = (\operatorname{Re} ,\operatorname{Im} )({z^n})$$
where, with the origin, z = 0, within the interior of Ω, n is restricted to non-negative values. So we have
$$\phi = \operatorname{Re} \sum\limits_0^\infty {{c_n}{z^n}}$$
for appropriate complex numbers { c n }. As an approximation we sum to some N -1 and determine the 2N real parameters by calculating Ø on 2N suitably chosen points on ∂Ω. Presumably, end of problem.


Refraction Lution Ghost 


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© Springer Science+Business Media New York 1994

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  • Mitchell J. Feigenbaum

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