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14 Legendre and Related FunctionsReal Arguments

§14.2 Differential Equations

Contents
  1. §14.2(i) Legendre’s Equation
  2. §14.2(ii) Associated Legendre Equation
  3. §14.2(iii) Numerically Satisfactory Solutions
  4. §14.2(iv) Wronskians and Cross-Products

§14.2(i) Legendre’s Equation

14.2.1 (1x2)d2wdx22xdwdx+ν(ν+1)w=0.

Standard solutions: 𝖯ν(±x), 𝖰ν(±x), 𝖰ν1(±x), Pν(±x), Qν(±x), Qν1(±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 (1x2)d2wdx22xdwdx+(ν(ν+1)μ21x2)w=0.

Standard solutions: 𝖯νμ(±x), 𝖯νμ(±x), 𝖰νμ(±x), 𝖰ν1μ(±x), Pνμ(±x), Pνμ(±x), 𝑸νμ(±x), 𝑸ν1μ(±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), 𝖯12+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 μν0,1,2,, and μ+ν1,2,3,, 𝖯νμ(x) and 𝖯νμ(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 μν=0,1,2,, or μ+ν=1,2,3,, 𝖯νμ(x) and 𝖯νμ(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 ν12, Pνμ(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, Pνμ(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{𝖯νμ(x),𝖯νμ(x)}=2Γ(μν)Γ(ν+μ+1)(1x2),
14.2.4 𝒲{𝖯νμ(x),𝖰νμ(x)}=Γ(ν+μ+1)Γ(νμ+1)(1x2),
14.2.5 𝖯ν+1μ(x)𝖰νμ(x)𝖯νμ(x)𝖰ν+1μ(x)=Γ(ν+μ+1)Γ(νμ+2),
14.2.6 𝒲{𝖯νμ(x),𝖰νμ(x)} =cos(μπ)1x2,
14.2.7 𝒲{Pνμ(x),Pνμ(x)} =𝒲{𝖯νμ(x),𝖯νμ(x)}=2sin(μπ)π(1x2),
14.2.8 𝒲{Pνμ(x),𝑸νμ(x)}=1Γ(ν+μ+1)(x21),
14.2.9 𝒲{𝑸νμ(x),𝑸ν1μ(x)}=cos(νπ)x21,
14.2.10 𝒲{Pνμ(x),Qνμ(x)}=eμπiΓ(ν+μ+1)Γ(νμ+1)(x21),
14.2.11 Pν+1μ(x)Qνμ(x)Pνμ(x)Qν+1μ(x)=eμπiΓ(ν+μ+1)Γ(νμ+2).