Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)).
The conical functions 
appear in
boundary-value problems for the Laplace equation in toroidal coordinates
(§14.19(i)) for regions bounded by cones, by two intersecting
spheres, or by one or two confocal hyperboloids of revolution
(Kölbig (1981)). These functions are also used in the Mehler–Fock
integral transform (§14.20(vi)) for problems in potential and heat
theory, and in elementary particle physics (Sneddon (1972, Chapter 7)
and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform
generalize to Jacobi functions and the Jacobi transform; see
Koornwinder (1984a) and references therein.
Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). See also §18.39.
Legendre functions 
of complex degree 
appear in the application of complex angular momentum
techniques to atomic and molecular scattering (Connor and Mackay (1979)).
