A direct method for modifying certain phase-integral approximations of arbitrary order

✍ Scribed by Nanny Fröman; Per Olof Fröman

• Publisher: Elsevier Science
• Year: 1974
• Tongue: English
• Weight: 248 KB
• Volume: 83
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(74)90379-0

✦ Synopsis

The approximate solution of the differential equation d'+d? + Q'(z)+ = 0 by a general modification of certain phase-integral approximations of arbitrary order is considered. A consistent modification of these higher-order phase-integral approximations is derived on the assumption that one has found a function Q&,(z) which makes the modified first-order approximation good at certain singular points of Q+), where the unmodified approximation would break down.

I. I NTRODU~TION

In recent years considerable interest has been focused on modifications of the higher-order JWKB-approximations in order to master the difficulties which often appear at the origin when these approximations are applied to radial problems [l-5]. In the present paper we shall be concerned with certain phase-integral approximations [6, 71, the higher-orders of which are related to, but not identical to the higher-order JWKB-approximations. Our aim is to expound a conceptually simple modification procedure which can be used quite generally (not only in radial problems) for making the phase-integral approximations useful at certain singular points where they would otherwise fail. The expressions obtained for the modified phase-integral approximations of any order are written in a comparatively simple form and can be given explicitly for all orders of approximation that are ever likely to be of interest in practice for solving physical problems [S, 93.