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14 Legendre and Related FunctionsReal Arguments

§14.17 Integrals

Contents
  1. §14.17(i) Indefinite Integrals
  2. §14.17(ii) Barnes’ Integral
  3. §14.17(iii) Orthogonality Properties
  4. §14.17(iv) Definite Integrals of Products
  5. §14.17(v) Laplace Transforms
  6. §14.17(vi) Mellin Transforms

§14.17(i) Indefinite Integrals

14.17.1 (1x2)μ/2𝖯νμ(x)dx=(1x2)(μ1)/2𝖯νμ1(x).
14.17.2 (1x2)μ/2𝖯νμ(x)dx=(1x2)(μ+1)/2(νμ)(ν+μ+1)𝖯νμ+1(x),
μν or ν1.
14.17.3 x𝖯νμ(x)𝖰νμ(x)dx=12ν(ν+1)((μ2(ν+1)(ν+x2))𝖯νμ(x)𝖰νμ(x)+(ν+1)(νμ+1)x(𝖯νμ(x)𝖰ν+1μ(x)+𝖯ν+1μ(x)𝖰νμ(x))(νμ+1)2𝖯ν+1μ(x)𝖰ν+1μ(x)),
ν0,1.
14.17.4 x(1x2)3/2𝖯νμ(x)𝖰νμ(x)dx=1(14μ2)(1x2)1/2×((12μ2+2ν(ν+1))𝖯νμ(x)𝖰νμ(x)+(2ν+1)(μν1)x×(𝖯νμ(x)𝖰ν+1μ(x)+𝖯ν+1μ(x)𝖰νμ(x))+2(μν1)2𝖯ν+1μ(x)𝖰ν+1μ(x)),
μ±12.

In (14.17.1)–(14.17.4), 𝖯 may be replaced by 𝖰, and in (14.17.3) and (14.17.4), 𝖰 may be replaced by 𝖯.

For further results, see Maximon (1955) and Prudnikov et al. (1990, pp. 37–39). See also (14.12.2), (14.12.5), and (14.12.12).

§14.17(ii) Barnes’ Integral

14.17.5 01xσ(1x2)μ/2𝖯νμ(x)dx=Γ(12σ+12)Γ(12σ+1)2μ+1Γ(12σ12ν+12μ+1)Γ(12σ+12ν+12μ+32),
σ>1, μ>1.

§14.17(iii) Orthogonality Properties

For l,m,n=0,1,2,,

14.17.6 11𝖯lm(x)𝖯nm(x)dx=(n+m)!(nm)!(n+12)δl,n,
14.17.7 11𝖯lm(x)𝖯nm(x)dx =(1)ml+12δl,n,
14.17.8 11𝖯nl(x)𝖯nm(x)1x2dx =(n+m)!(nm)!mδl,m,
m>0,
14.17.9 11𝖯nl(x)𝖯nm(x)1x2dx =(1)llδl,m,
l>0.

Orthogonality relations for the associated Legendre functions of imaginary order are given in Bielski (2013).

§14.17(iv) Definite Integrals of Products

With ψ(x)=Γ(x)/Γ(x)5.2(i)),

14.17.10 11𝖯ν(x)𝖯λ(x)dx=2(2sin(νπ)sin(λπ)(ψ(ν+1)ψ(λ+1))+πsin((λν)π))π2(λν)(λ+ν+1),
λν or ν1.
14.17.11 11(𝖯ν(x))2dx=π22sin2(νπ)ψ(ν+1)π2(ν+12),
ν12.
14.17.12 11𝖰ν(x)𝖰λ(x)dx=((ψ(ν+1)ψ(λ+1))(1+cos(νπ)cos(λπ))+12πsin((λν)π))(λν)(λ+ν+1),
λν or ν1, λ and ν1,2,3,.
14.17.13 11(𝖰ν(x))2dx=π22(1+cos2(νπ))ψ(ν+1)2(2ν+1),
ν12 or 1,2,3,.
14.17.14 11𝖯ν(x)𝖰λ(x)dx=2sin(νπ)cos(λπ)(ψ(ν+1)ψ(λ+1))+πcos((λν)π)ππ(λν)(λ+ν+1),
λ>0, ν>0, λν.
14.17.15 11𝖯ν(x)𝖰ν(x)dx=sin(2νπ)ψ(ν+1)π(2ν+1),
ν>0.
14.17.16 11𝖯lm(x)𝖰nm(x)dx=(1(1)l+n)(l+m)!(ln)(l+n+1)(lm)!,
l,m,n=0,1,2,, ln.
14.17.17 0π𝖰l(cosθ)𝖯m(cosθ)𝖯n(cosθ)sinθdθ=0,
l,m,n=1,2,3,, |mn|<l<m+n.

(When l+m+n is even the condition |mn|<l<m+n is not needed.) Next,

14.17.18 1Pν(x)Qλ(x)dx=1(λν)(ν+λ+1),
λ>ν>0.
14.17.19 1Qν(x)Qλ(x)dx=ψ(λ+1)ψ(ν+1)(λν)(λ+ν+1),
(λ+ν)>1, λν, λ and ν1,2,3,.
14.17.20 1(Qν(x))2dx=ψ(ν+1)2ν+1,
ν>12.

For further results, see Prudnikov et al. (1990, pp. 194–240); also (34.3.21).

§14.17(v) Laplace Transforms

For Laplace transforms and inverse Laplace transforms involving associated Legendre functions, see Erdélyi et al. (1954a, pp. 179–181, 270–272), Oberhettinger and Badii (1973, pp. 113–118, 317–324), Prudnikov et al. (1992a, §§3.22, 3.32, and 3.33), and Prudnikov et al. (1992b, §§3.20, 3.30, and 3.31).

§14.17(vi) Mellin Transforms

For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).