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Smoothing of the Stokes Phenomenon for High-Order Differential Equations

R. B. Paris
Proceedings: Mathematical and Physical Sciences
Vol. 436, No. 1896 (Jan. 8, 1992), pp. 165-186
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52027
Page Count: 22
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Smoothing of the Stokes Phenomenon for High-Order Differential Equations
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Abstract

We extend the class of functions for which the smooth transition of a Stokes multiplier across a Stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n. The equation chosen admits solutions of hypergeometric function type which, in the case n = 2, are related to the parabolic cylinder functions. In general, the solutions of this equation involve compound asymptotic expansions, valid in certain sectors of the complex z-plane, with more than one dominant and subdominant series. The functional form of the Stokes multipliers, expressed in terms of an appropriately scaled variable describing transition across a Stokes line, is found to obey the error function smoothing law derived by Berry.

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