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
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
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.
- George F. Carrier, Max Krook, and Carl E. Pearson, Functions of a complex variable: Theory and technique, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222256
- Selected numerical methods for linear equations, polynomial equations, partial differential equations, conformal mappings, Regnecentralen, Copenhagen, 1962. Report on studies sponsored by The Carlsberg Foundation. MR 0149638
- Dieter Gaier, Konstruktive Methoden der konformen Abbildung, Springer Tracts in Natural Philosophy, Vol. 3, Springer-Verlag, Berlin, 1964 (German). MR 0199360
- Peter Henrici, Fast Fourier methods in computational complex analysis, SIAM Rev. 21 (1979), no. 4, 481–527. MR 545882, DOI https://doi.org/10.1137/1021093
- Daniel I. Meiron, Steven A. Orszag, and Moshe Israeli, Applications of numerical conformal mapping, J. Comput. Phys. 40 (1981), no. 2, 345–360. MR 617103, DOI https://doi.org/10.1016/0021-9991%2881%2990215-1
- Bengt Fornberg, A numerical method for conformal mappings, SIAM J. Sci. Statist. Comput. 1 (1980), no. 3, 386–400. MR 596032, DOI https://doi.org/10.1137/0901027
W. Murray (ed.), Numerical methods for unconstrained optimization, Academic Press (1972)
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a complex variable, McGraw-Hill, 1966
Chr. Anderson et al., Conformal mapping (Chap. III) Selected numerical methods, ed. Christian Gram, Regnecentralen, Copenhagen, 1962
D. Gaier, Konstructive Methoden der konformen Abbilding, Springer, 1964
P. Henrici, Fast Fourier methods in computational complex analysis, SIAM Review 21, 481–527 (1979)
D. I. Meiron and S. A. Orszag, Applications of numerical conformal mapping, J. Comp. Phys. 40, 345–359 (1981)
B. Fornberg, A numerical method for conformal mappings, SIAM J. Sci. Stat. Comp. 1, 386–400 (1980)
W. Murray (ed.), Numerical methods for unconstrained optimization, Academic Press (1972)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
30C30,
30-04,
30B40
Retrieve articles in all journals
with MSC:
30C30,
30-04,
30B40
Additional Information
Article copyright:
© Copyright 1983
American Mathematical Society