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Liouville-Green expansions of exponential form, with an application to modified Bessel functions

Part of: Asymptotic theory

Published online by Cambridge University Press:  29 January 2019

T. M. Dunster
T. M. Dunster*
Affiliation:
Department of Mathematics and Statistics, San Diego State University, San Diego, CA92182-7720, USA ([email protected])
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Abstract

Linear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

Keywords

WKB methodsasymptotic expansionsBessel functions

MSC classification

Primary: 34E20: Singular perturbations, turning point theory, WKB methods
Secondary: 34E05: Asymptotic expansions 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1289 - 1311
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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