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Computing toroidal functions for wide ranges of the parameters. (English) Zbl 0994.65018
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.
65D20Computation of special functions, construction of tables
33C47Other special orthogonal polynomials and functions