Computation of Jacobi functions of the second kind for use in nearside-farside scattering theory
✍ Scribed by Jet Wimp; P. McCabe; J.N.L. Connor
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 944 KB
- Volume
- 82
- Category
- Article
- ISSN
- 0377-0427
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✦ Synopsis
The nearside-farside decomposition of a partial wave series is currently being used to understand the angular scattering of atom-diatom collision systems. In this theory, it is necessary to compute Jacobi functions of the second kind on the cut. These functions are denoted by Q~'~(cos 0), where n, ~, r, may be large positive integers. The Q~'~(cos 0) can be computed from a three-term linear recurrence relation provided the initial values corresponding to n = 0 and 1, are known. We derive explicit formulas for Q~o ~' ~1 (cos 0), Q~'~)(cos 0) in terms of elementary transcendental functions. A new generating function for Jacobi functions of nonintegral degree off the cut is obtained, a special case of which yields a generating function for Q~'~)(cos 0). This is used to check the numerical results, as is a Casoratian relation. We show that the recurrence for Qt~.~)(cos 0) is stable in the forward direction with errors growing like O (n). We also present some numerics demonstrating the success of the method.