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Associated Legendre functions of the first and second kind,
and
are in general characterized by
three complex constants: the degree
the order
and
the argument z. Toroidal (or ring) functions are
the associated Legendre functions with odd-half-integer degree and
integer order. Here we illustrate some of the striking symmetric
characteristics of toroidal functions. We then show how these lead
to an alternative azimuthal Fourier series representation of the
Green's function for the rotational Laplace systems whose coefficients
are given in terms of toroidal functions of the first kind.