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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 9 Number 1 October 2023

A Note on the Approximate Formula for the Associated Legendre Function


Authors: Mitsuru Yoshida
Pages: 1-4
DOI: doi.org/10.32861/ajams.91.1.4
Abstract
The approximate formula for the associated Legendre function has been used for the determination of the wavelength and phase velocity of free oscillations of the Earth and long-period surface waves. The approximate formula is defined for several mathematical parameters. One of them is n >>1/ε, where n is the angular degree and εis a parameter related to the colatitude angle θ. In the present study  the relationship between n and ε is defined as n0 = [ 1/ε ] +1, where the notation [ ] denotes the Gaussian symbol.  If a condition n ≧ n0   (n>1/ε ) is assumed, the approximate formula may be reasonably applied for the angular degrees n ≧2 for π/6≦θ≦5π/6. This condition is in harmony with the practical usage implemented conventionally.