A Note on the Approximate Formula for the Associated Legendre Function

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.