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A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

Published online by Cambridge University Press:  14 November 2011

B. J. Harris
B. J. Harris
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115-2888, U.S.A.
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Synopsis

In an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equation

has the asymptotic expansion

as |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.

We show that if the real valued function q admits the expansion

in a neighbourhood of 0, then

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 102 , Issue 3-4 , 1986 , pp. 253 - 257
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Atkinson, F. V.. On the asymptotic behaviour of the Titchmarsh–Weyl m-coefficients and the spectral functions for scalar second order differential expressions. Lecture Notes in Mathematics 949. (Berlin: Springer, 1982).Google Scholar
2Bennewitz, C. and Everitt, W. A.. Some remarks on the Titchmarsh–Weyl m-coefficient. In Tribute to Åke Pleijel (Uppsala: Department of Mathematics, University of Uppsala, 1980).Google Scholar
3Erdélyi., A.Asymptotic expansions (New York: Dover, 1956).Google Scholar
4Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function. J. London Math. Soc. (2) 30 (1984), 110–118.CrossRefGoogle Scholar
5Harris, B. J.. The asymptotic form of the spectral functions associated with a class of Sturm–Liouville equations. Proc. Roy. Soc. Edinburgh Sect. A 100 (1985), 343–360.Google Scholar
6Harris, B. J.. The asymptotic form of the Titchmarsh-Weyl m-function for second order linear differential equations with analytic coefficient. J. Differential Equations (to appear).Google Scholar
7Titchmarsh, E. C.. Eigenfunction expansions associated with second order differential equations Vol. 1. Oxford: Oxford University Press, 2nd edition, 1962).Google Scholar
8Weyl, H.. Uber gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklungen willkurlicher funktionen. Math. Ann. 68 (1910), 220–269.Google Scholar