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Riemann Maps and World Maps

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

Abstract

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),
$$\Delta {\psi _n} = 0 \Rightarrow {\psi _n} = (\operatorname{Re} ,\operatorname{Im} )({z^n})$$
(1)
where, with the origin, z = 0, within the interior of Ω, n is restricted to non-negative values. So we have
$$\phi = \operatorname{Re} \sum\limits_0^\infty {{c_n}{z^n}}$$
(2)
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.

Keywords

Subharmonic Function Linear Invariance Worth Bearing Deep Invagination Rational Function Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1994

Authors and Affiliations

  • Mitchell J. Feigenbaum

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