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14 Legendre and Related FunctionsReal Arguments

§14.5 Special Values

Contents
  1. §14.5(i) x=0
  2. §14.5(ii) μ=0, ν=0,1
  3. §14.5(iii) μ=±12
  4. §14.5(iv) μ=ν
  5. §14.5(v) μ=0, ν=±12
  6. §14.5(vi) Addendum to §14.5(ii)μ=0, ν=2

§14.5(i) x=0

14.5.1 𝖯νμ(0)=2μπ1/2Γ(12ν12μ+1)Γ(1212ν12μ),
14.5.2 d𝖯νμ(x)dx|x=0=2μ+1π1/2Γ(12ν12μ+12)Γ(12ν12μ),
14.5.3 𝖰νμ(0)=2μ1π1/2sin(12(ν+μ)π)Γ(12ν+12μ+12)Γ(12ν12μ+1),
ν+μ1,2,3,,
14.5.4 d𝖰νμ(x)dx|x=0=2μπ1/2cos(12(ν+μ)π)Γ(12ν+12μ+1)Γ(12ν12μ+12),
ν+μ1,2,3,.

§14.5(ii) μ=0, ν=0,1

14.5.5 𝖯0(x)=P0(x)=1,
14.5.6 𝖯1(x)=P1(x)=x.
14.5.7 𝖰0(x) =12ln(1+x1x),
14.5.8 𝖰1(x) =x2ln(1+x1x)1.
14.5.9 𝑸0(x) =12ln(x+1x1),
14.5.10 𝑸1(x) =x2ln(x+1x1)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 𝖯ν1/2(cosθ) =(2πsinθ)1/2sin((ν+12)θ)ν+12,
14.5.13 𝖰ν1/2(cosθ) =(π2sinθ)1/2sin((ν+12)θ),
14.5.14 𝖰ν1/2(cosθ)=(π2sinθ)1/2cos((ν+12)θ)ν+12.
14.5.15 Pν1/2(coshξ) =(2πsinhξ)1/2cosh((ν+12)ξ),
14.5.16 Pν1/2(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

14.5.18 𝖯νν(cosθ) =(sinθ)ν2νΓ(ν+1),
14.5.19 Pνν(coshξ) =(sinhξ)ν2νΓ(ν+1).

§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 𝖯12(cosθ) =2πK(sin(12θ)),
14.5.22 𝖰12(cosθ) =K(cos(12θ))2E(cos(12θ)),
14.5.23 𝖰12(cosθ) =K(cos(12θ)).
14.5.24 P12(coshξ) =2πeξ/2E((1e2ξ)1/2),
14.5.25 P12(coshξ) =2πcosh(12ξ)K(tanh(12ξ)),
14.5.26 𝑸12(coshξ)=2π1/2coshξsech(12ξ)K(sech(12ξ))4π1/2cosh(12ξ)E(sech(12ξ)),
14.5.27 𝑸12(coshξ)=2π1/2eξ/2K(eξ).

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

14.5.28 𝖯2(x) =P2(x)=3x212,
14.5.29 𝖰2(x) =3x214ln(1+x1x)32x,
14.5.30 𝑸2(x) =3x218ln(x+1x1)34x.