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14 Legendre and Related FunctionsReal Arguments

§14.2 Differential Equations

Contents

§14.2(i) Legendre’s Equation

14.2.1 (1-x2)d2wdx2-2xdwdx+ν(ν+1)w=0.

Standard solutions: Pν(±x), Qν(±x), Q-ν-1(±x), Pν(±x), Qν(±x), Q-ν-1(±x). Pν(x) and Qν(x) are real when ν and x(-1,1), and Pν(x) and Qν(x) are real when ν and x(1,).

§14.2(ii) Associated Legendre Equation

14.2.2 (1-x2)d2wdx2-2xdwdx+(ν(ν+1)-μ21-x2)w=0.

Standard solutions: Pνμ(±x), Pν-μ(±x), Qνμ(±x), Q-ν-1μ(±x), Pνμ(±x), Pν-μ(±x), Qνμ(±x), Q-ν-1μ(±x).

(14.2.2) reduces to (14.2.1) when μ=0. Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations Pν0(x)=Pν(x), Qν0(x)=Qν(x), Pν0(x)=Pν(x), Qν0(x)=Qν(x), Qν0(x)=Qν(x)=Qν(x)/Γ(ν+1).

Pνμ(x), P-12+iτμ(x), and Qνμ(x) are real when ν, μ, and τ, and x(-1,1); Pνμ(x) and Qνμ(x) are real when ν and μ, and x(1,).

Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions Pνμ(x) and Qνμ(x) lie in the interval (-1,1), and the arguments of the functions Pνμ(x), Qνμ(x), and Qνμ(x) lie in the interval (1,). For extensions to complex arguments see §§14.2114.28.

§14.2(iii) Numerically Satisfactory Solutions

Equation (14.2.2) has regular singularities at x=1, -1, and , with exponent pairs {-12μ,12μ}, {-12μ,12μ}, and {-ν-1,ν}, respectively; compare §2.7(i).

When μ-ν0,-1,-2,, and μ+ν-1,-2,-3,, Pν-μ(x) and Pν-μ(-x) are linearly independent, and when μ0 they are recessive at x=1 and x=-1, respectively. Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval -1<x<1. When μ-ν=0,-1,-2,, or μ+ν=-1,-2,-3,, Pν-μ(x) and Pν-μ(-x) are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

When μ0 and ν-12, Pν-μ(x) and Qνμ(x) are linearly independent, and recessive at x=1 and x=, respectively. Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval 1<x<. With the same conditions, Pν-μ(-x) and Qνμ(-x) comprise a numerically satisfactory pair of solutions in the interval -<x<-1.

§14.2(iv) Wronskians and Cross-Products

14.2.3 𝒲{Pν-μ(x),Pν-μ(-x)}=2Γ(μ-ν)Γ(ν+μ+1)(1-x2),
14.2.4 𝒲{Pνμ(x),Qνμ(x)}=Γ(ν+μ+1)Γ(ν-μ+1)(1-x2),
14.2.5 Pν+1μ(x)Qνμ(x)-Pνμ(x)Qν+1μ(x)=Γ(ν+μ+1)Γ(ν-μ+2),
14.2.6 𝒲{Pν-μ(x),Qνμ(x)} =cos(μπ)1-x2,
14.2.7 𝒲{Pν-μ(x),Pνμ(x)} =𝒲{Pν-μ(x),Pνμ(x)}
=2sin(μπ)π(1-x2),
14.2.8 𝒲{Pν-μ(x),Qνμ(x)}=-1Γ(ν+μ+1)(x2-1),
14.2.9 𝒲{Qνμ(x),Q-ν-1μ(x)}=cos(νπ)x2-1,
14.2.10 𝒲{Pνμ(x),Qνμ(x)}=-eμπiΓ(ν+μ+1)Γ(ν-μ+1)(x2-1),
14.2.11 Pν+1μ(x)Qνμ(x)-Pνμ(x)Qν+1μ(x)=eμπiΓ(ν+μ+1)Γ(ν-μ+2).