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Error bounds for a uniform asymptotic expansion of the Legendre function P n -m (coshz). (English) Zbl 0655.33004
The Legendre function P n -m (cohz) is considered for large values of n. The asymptotic expansion is in terms of the modified Bessel function I m (z), and holds uniformly with respect to z[0,); the parameter m is fixed, m>-1/2. The method is based on an integral representation of the Legendre function. A recurrence relation is derived for the coefficients in the expansion, and computable error bounds are derived for the remainders. A comparison is given of the new expansion with an earlier expansion given by Olver, especially with respect to the bounds for the remainders.
Reviewer: N.M.Temme
33C55Spherical harmonics
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)