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Academic Journal of Applied Mathematical Sciences

Online ISSN: 2415-2188
Print ISSN: 2415-5225

Quarterly Published (4 Issues Per Year)





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Volume 11 Number 1 April 2025

A Classification of the Mathematical Expressions of the Asymptotic Series Expansion of the Associated Legendre Function


Authors: Mitsuru Yoshida
Pages: 1-7
DOI: doi.org/10.32861/ajams.111.1.7
Abstract
The mathematical expressions of the asymptotic series expansion of  the associated Legendre function, which  have been widely  described in the mathematical  literature, are here shown to be classifiable into two types according to their manner of  expression. The Type-A mathematical expressions are represented by the first three higher terms of the asymptotic series, while the Type-B expressions are represented by the infinite sequence. As a result,  the Type-B mathematical expressions differ slightly from one another, despite having identical computation results. The distinct representation  of the Type-B expressions arises from (1) the treatment of the gamma functions including the colatitude angle in the spherical coordinate system and the azimuthal order number of the associated Legendre function and (2) the method of expressing the phase angle term. Moreover, one of the mathematical expressions of the Type-B is derived using a  hypergeometric formula different from other usual one.