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Conformal maps in methods of asymptotic continuation

Published online by Cambridge University Press:  14 November 2011

H. Gingold
H. Gingold
Affiliation:
Department of Mathematics, West Virginia University, College of Arts and Sciences, Morgantown, West Virginia 26506, U.S.A.
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Synopsis

Take the coefficients of a Taylor series expansion of a holomorphic function about its regular point zR. It is known that the holomorphic function possesses an asymptotic expansion about a possibly singular point zs. We show how to construct and calculate the coefficients in the asymptotic expansion from the coefficient of the Taylor series. The main theorem demonstrates that a suitable conformal map is a decisive step in dealing with the problem above. Therefore, a suitable conformal map is critical to a successful summation of divergent series. Some other methods which utilise orthogonal polynomial and Cesaro summability are also discussed. The paper may serve as a theoretical basis for a new computational method.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 319 - 335
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Bieberbach, L.. Lehrbach Der Funktionentheorie (Leipzig and Berlin: B. G. Teubner, 1930).Google Scholar
2Dienes, P.. The Taylor Series (New York: Dover, 1957).Google Scholar
3Ford, Walter Burton. Studies on Divergent Series and Summability and the Asymptotic Developments of Functions Defined by Maclaurin Series (New York: Chelsea, 1960).Google Scholar
4Gingold, H.. A new basis for singularly perturbed problems. A representation theorem. Adv. in Appl. Math. 1 (1980), 67–107.CrossRefGoogle Scholar
5Gingold, H.. Problems of mathematical physics which depend on a parameter and Jacobi series. Adv. in Appl. Math. 15 (1984), 357–397.CrossRefGoogle Scholar
6Gould, H.. Coefficient identities for powers of Taylor and Dirichlet series. Amer. Math. Monthly 81 (1974), 3–14.CrossRefGoogle Scholar
7Hardy, Godfrey Harold. Divergent Series (Oxford: Oxford University Press, 1949).Google Scholar
8Hille, E.. Analytic Function Theory, Vol. II (New York: Blaisdell, 1963).Google Scholar
9Knopp, Konrad. Theory and Application of Infinite Series, trans, from the 2nd German edn. (New York: Hafner, 1971).Google Scholar
10Kober, H.. Dictionary of Conformal Representations (New York: Dover, 1957).Google Scholar
11Maddox, Ivor John. Infinite Matrices of Operators (Berlin: Springer, 1980).CrossRefGoogle Scholar
12Natanson, I. P.. Constructive Function Theory, translated by Schulenberger, J. R.. Vol. II (New York: Ungar, 1965).Google Scholar
13Nehari, Z.. Conformal Mapping (New York: McGraw-Hill, 1952).Google Scholar
14Peyerimhoff, Alexander. Lectures on Summability (Berlin: Springer, 1969).CrossRefGoogle Scholar
15Poincaré, H.. Sur les integrales irrfigulieres des équations linéares. Ada Math. 8 (1886), 295–344.Google Scholar
16Poincaré, H.New Methods of Celestial Mechanics, Vol. II (Washington, D.C.: NASA Technical Translation, 1967).Google Scholar
17Szegö, G.. Orthogonal Polynomials, Colloquium Publications 23 (Providence, R.I.: American Mathematical Society, 1959).Google Scholar
18Dyke, M. Van. Perturbation Methods in Fluid Mechanics (London/New York: Academic Press, 1964).Google Scholar
19Wasow, W. R.. Asymptotic Expansions for Ordinary Differential Equations (New York: Wiley, 1965).Google Scholar
20Witham, G. B.. Linear and Nonlinear Waves (New York: Wiley, 1974).Google Scholar
21Zamansky, Marc. La Summation des Series Divergentes (Paris: Gauthier–Villars, 1954).Google Scholar
22Zeller, Karl. Theorie Der Limitierungsverfahren (Berlin: Springer, 1958).CrossRefGoogle Scholar
23Zygmund, A.. Trigonometric Series, 2nd edn., Vol. I (New York: Cambridge University Press, 1959).Google Scholar
24Baker, G. A. Jr.. and Graves-Morris, P.. Pade Approximations, Vols. I, II (Reading, Massachusetts: Addison-Wesley, 1981).Google Scholar
25Brent, R. P.. and Kung, H. T.. algorithms for composition and reversion of power series. In Analytic Computational Complexity, ed. Traub, J. F.. pp. 217–225 (New York: Academic Press, 1976).CrossRefGoogle Scholar
26Gingold, H.. Numerical aspects of a recent theorem regarding conformal maps (in prep.).Google Scholar
27Kapenga, A. J.. Jacobi Moments in Applied Mathematics with Computer Applications (Ph.D. Dissertation, Western Michigan University, Kalamazoo, Michigan, 1985).Google Scholar
28Dyke, M. Van. Computed-extended series. Ann. Rev. Fluid Mech. 16 (1984), 287–309.CrossRefGoogle Scholar