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14 Legendre and Related FunctionsReal Arguments

§14.12 Integral Representations

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§14.12(i) -1<x<1

Mehler–Dirichlet Formula

14.12.1 Pνμ(cosθ) =21/2(sinθ)μπ1/2Γ(12-μ)0θcos((ν+12)t)(cost-cosθ)μ+(1/2)dt,
0<θ<π, μ<12.
14.12.2 Pν-μ(x) =(1-x2)-μ/2Γ(μ)x1Pν(t)(t-x)μ-1dt,
μ>0;

compare (14.6.6).

14.12.3 Qνμ(cosθ)=π1/2Γ(ν+μ+1)(sinθ)μ2μ+1Γ(μ+12)Γ(ν-μ+1)(0(sinht)2μ(cosθ+isinθcosht)ν+μ+1dt+0(sinht)2μ(cosθ-isinθcosht)ν+μ+1dt,)
0<θ<π, μ>-12, (ν±μ)>-1.

§14.12(ii) 1<x<

14.12.4 Pν-μ(x) =21/2Γ(μ+12)(x2-1)μ/2π1/2Γ(ν+μ+1)Γ(μ-ν)0cosh((ν+12)t)(x+cosht)μ+(1/2)dt,
ν+μ-1,-2,-3,, (μ-ν)>0.
14.12.5 Pν-μ(x) =(x2-1)-μ/2Γ(μ)1xPν(t)(x-t)μ-1dt,
μ>0.
14.12.6 Qνμ(x) =π1/2(x2-1)μ/22μΓ(μ+12)Γ(ν-μ+1)0(sinht)2μ(x+(x2-1)1/2cosht)ν+μ+1dt,
(ν+1)>μ>-12.
14.12.7 Pνm(x) =(ν+1)mπ0π(x+(x2-1)1/2cosϕ)νcos(mϕ)dϕ,
14.12.8 Pnm(x) =2mm!(n+m)!(x2-1)m/2(2m)!(n-m)!π0π(x+(x2-1)1/2cosϕ)n-m(sinϕ)2mdϕ,
nm.
14.12.9 Qnm(x)=1n!0u(x-(x2-1)1/2cosht)ncosh(mt)dt,

where

14.12.10 u=12ln(x+1x-1).
14.12.11 Qnm(x)=(x2-1)m/22n+1n!-11(1-t2)n(x-t)n+m+1dt,
14.12.12 Qnm(x)=1(n-m)!Pnm(x)xdt(t2-1)(Pnm(t))2,
nm.

Neumann’s Integral

Heine’s Integral

For further integral representations see Erdélyi et al. (1953a, pp. 158–159) and Magnus et al. (1966, pp. 184–190), and for contour integrals and other representations see §14.25.