Legendre functions with both parameters large

Author:

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 ∆ are arbitrary fixed numbers such that 0 < δ < 1 and ∆ > 0. The values of m and n + ½ are either both real, or both purely imaginary. In all cases explicit bounds are supplied for the error terms associated with the approximations.

Publisher

The Royal Society

Subject

General Engineering

Reference7 articles.

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2. Classical limits of clebsch-gordan coefficients, racah coefficients and Dmnl (φ, ϑ, ψ)-functions

3. M iller J . G. P. 1955 Tables o f Weberparabolic cylinderfunctions. L o n d o n : H er M ajesty's Stationery Office.

4. Olver F. W. J . 1974 Asymptotics and special functions. L o n d o n : Academic Press.

5. Second-order linear differential equations with two turning points

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