Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Numerical conformal mapping and analytic continuation


Author: Frederic Bisshopp
Journal: Quart. Appl. Math. 41 (1983), 125-142
MSC: Primary 30C30; Secondary 30-04, 30B40
DOI: https://doi.org/10.1090/qam/700667
MathSciNet review: 700667
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Abstract: A numerical method for determination of least-square approximations of an arbitrary complex mapping function is derived here and implemented with fast Fourier transforms (FFTs). An essential feature of the method is the factoring of a discrete Hilbert transform in a pair of Fourier transforms in order to reduce the operation count of the longest computation to $ O\left( {N\log N} \right)$. A similar factoring of the discrete Poisson integral formula allows an explicit inversion of it in $ O\left( {N\log N} \right)$ operations instead of $ O\left( {{N^3}} \right)$(N$ ^{3}$). The resulting scheme for analytic continuation appears to be considerably more reliable than the evaluation of polynomials. Examples are treated, and APL implementations of algorithms are provided.


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DOI: https://doi.org/10.1090/qam/700667
Article copyright: © Copyright 1983 American Mathematical Society


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