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Legendre Functions with Both Parameters Large

F. W. J. Olver
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 278, No. 1279 (Mar. 20, 1975), pp. 175-185
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/74500
Page Count: 11
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Legendre Functions with Both Parameters Large
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Abstract

By application of the theory for second-order linear differential equations with two turning points developed in the preceding paper, some new asymptotic approximations are obtained for the associated Legendre functions when both the degree n and order m are large. The approximations are expressed in terms of parabolic cylinder functions, and are uniformly valid with respect to x ∈ (-1, 1) and m/(n + 1/2) ∈ [δ , 1 + Δ ], where δ and Δ are arbitrary fixed numbers such that 0 < δ < 1 and Δ > 0. The values of m and n + 1/2 are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.

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