Alternate Forms of the Associated Legendre Functions for Use in Geomagnetic Modeling

Abstract

An inconvenience attending traditional use of associated Legendre functions in global modeling is that the functions are not separable with respect to the two indices (order and degree). In 1973 Merilees suggested a way to avoid the problem by showing that associated Legendre functions of order m and degree m+k can be expressed in terms of elementary functions as
P^m_m+k(θ)=sin^m(θ)∑^k_i=0a^m_kicos(iθ)
where a^m_ki, the constants to be determined, are somewhat analogous to Fourier coefficients. Merilees noted that there are several advantages to using this form, but he also raises a question of precision for degree and order greater than 25. This note calls attention to some possible gains in time savings and accuracy in geomagnetic modeling based upon this form. For this purpose, expansions of associated Legendre polynomials in terms of sines and cosines of multiple angles are displayed up to degree and order 10. Examples are also given explaining how some surface spherical harmonics can be transformed into true Fourier series for selected polar great circle paths.