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An orthonormalization method for the approximate conformal mapping of multiply-connected domains. (English) Zbl 0703.30005
The authors use a known orthogonalization method related to the Bergman kernel function to effectively carry out the conformal mapping of a finite domain G of connectivity \(N\geq 2\) onto a circular ring with N-2 concentric circular slits, or onto other standard slit domains. Inner products are transformed into line integrals, and these are evaluated by Gauß-Legendre formulae. As basis functions the powers \(z^ m\) \((m=0,1,2,...)\) and \((z-a_ j)^ m\) \((m=2,3,...)\) are used (‘monomial basis’) or singular functions added (‘augmented basis’) whose purpose is to speed up the rate of approximation to the mapping function. Seven test examples, all of degree 3 or 4, show that the method can be recommended and that the augmented basis should be used whenever the mapping function is not regular on \(\bar G.\)
Reviewer: D.Gaier

MSC:
30C30 Schwarz-Christoffel-type mappings
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