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14 Legendre and Related FunctionsReal Arguments

§14.5 Special Values

Contents

§14.5(i) x=0

14.5.1 Pνμ(0)=2μπ1/2Γ(12ν-12μ+1)Γ(12-12ν-12μ),
14.5.2 dPνμ(x)dx|x=0=-2μ+1π1/2Γ(12ν-12μ+12)Γ(-12ν-12μ),
14.5.3 Qνμ(0)=-2μ-1π1/2sin(12(ν+μ)π)Γ(12ν+12μ+12)Γ(12ν-12μ+1),
ν+μ-1,-3,-5,,
14.5.4 dQνμ(x)dx|x=0=2μπ1/2cos(12(ν+μ)π)Γ(12ν+12μ+1)Γ(12ν-12μ+12),
ν+μ-2,-4,-6,.

§14.5(ii) μ=0, ν=0,1, and 2

14.5.5 P0(x)=P0(x)=1,
14.5.6 P1(x)=P1(x)=x.
14.5.7 Q0(x) =12ln(1+x1-x),
14.5.8 Q1(x) =x2ln(1+x1-x)-1.
14.5.9 Q0(x) =12ln(x+1x-1),
14.5.10 Q1(x) =x2ln(x+1x-1)-1.

For the corresponding formulas when ν=2 see §14.5(vi).

§14.5(iii) μ=±12

In this subsection and the next two, 0<θ<π and ξ>0.

14.5.11 Pν1/2(cosθ) =(2πsinθ)1/2cos((ν+12)θ),
14.5.12 Pν-1/2(cosθ) =(2πsinθ)1/2sin((ν+12)θ)ν+12,
14.5.13 Qν1/2(cosθ) =-(π2sinθ)1/2sin((ν+12)θ),
14.5.14 Qν-1/2(cosθ)=-(π2sinθ)1/2cos((ν+12)θ)ν+12.
14.5.15 Pν1/2(coshξ) =(2πsinhξ)1/2cosh((ν+12)ξ),
14.5.16 Pν-1/2(coshξ) =(2πsinhξ)1/2sinh((ν+12)ξ)ν+12,
14.5.17 Qν±1/2(coshξ) =(π2sinhξ)1/2exp(-(ν+12)ξ)Γ(ν+32).

§14.5(iv) μ=-ν

§14.5(v) μ=0,ν=±12

In this subsection K(k) and E(k) denote the complete elliptic integrals of the first and second kinds; see §19.2(ii).

14.5.20 P12(cosθ)=2π(2E(sin(12θ))-K(sin(12θ))),
14.5.21 P-12(cosθ) =2πK(sin(12θ)),
14.5.22 Q12(cosθ) =K(cos(12θ))-2E(cos(12θ)),
14.5.23 Q-12(cosθ) =K(cos(12θ)).
14.5.24 P12(coshξ) =2πeξ/2E((1-e-2ξ)1/2),
14.5.25 P-12(coshξ) =2πcosh(12ξ)K(tanh(12ξ)),
14.5.26 Q12(coshξ)=2π-1/2coshξsech(12ξ)K(sech(12ξ))-4π-1/2cosh(12ξ)E(sech(12ξ)),
14.5.27 Q-12(coshξ)=2π-1/2e-ξ/2K(e-ξ).

§14.5(vi) Addendum to §14.5(ii) μ=0,ν=0,1, and 2

14.5.28 P2(x) =P2(x)
=3x2-12,
14.5.29 Q2(x) =3x2-14ln(1+x1-x)-32x,
14.5.30 Q2(x) =3x2-18ln(x+1x-1)-34x.