Computing Toroidal Functions for Wide Ranges of the Parameters

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 Pνm(x) (for large m), which fills the gap left by previous methods.