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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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