ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
Silverstone [Phys. Rev. Lett. 55, 2523–2526 (1985)] has expressed the opinion that the traditional version of the JWKB connection formulas associated with a classical turning point is incorrect and that the correct version follows from the Borel-based summability of the asymptotic expansions for the Airy functions. In the present paper we show that this assertion is incorrect. After the Borel summation of the asymptotic expressions for the Airy functions has actually been performed explicitly, there are no connection problems, and thus no connection formulas, but exact relations expressing the functions Ai(z) and Bi(z) in terms of the Whittaker function, although with two defects: Due to the derivation there appear false restrictions on arg z, and there is an artificial cut that coincides with the Stokes line along the real z-axis. We also show that these exact relations can be obtained more appropriately in a direct way with the aid of standard formulas available in handbooks, whereby the defects mentioned do not appear. On the other hand, when one uses truncated asymptotic series, the Stokes phenomenon gives rise to connection problems, and the connection formulas, which one uses for handling these connection problems, are inherently one-directional. This property of the connection formulas is demonstrated and discussed anew, in general terms as well as with special regard to the Airy functions. © 1998 American Institute of Physics.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.532517
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