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Zbl 0994.65018
Gil, Amparo; Segura, Javier; Temme, Nico M.
Computing toroidal functions for wide ranges of the parameters. (English)
[J] J. Comput. Phys. 161, No.1, 204-217 (2000). ISSN 0021-9991

Summary: Associated Legendre functions of half-odd degree and arguments larger than one, also known as toroidal harmonics, appear in the solution of Dirichlet problems with toroidal symmetry. It is shown how the use of series expansions, continued fractions, and uniform asymptotic expansions, together with the application of recurrence relations over degrees and orders, permits the evaluation of the whole set of toroidal functions for a wide range of arguments, orders, and degrees. In particular, we provide a suitable uniform asymptotic expansion for $P^m(x)$ (for large $m$), which fills the gap left by previous methods.

MSC 2000:: *65D20 Computation of special functions
33C47 Other special orthogonal polynomials and functions

Keywords: Legendre functions; toroidal harmonics; uniform asymptotics; Dirichlet problems; series expansions; continued fractions

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