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14 Legendre and Related FunctionsReal Arguments

§14.8 Behavior at Singularities

Contents
  1. §14.8(i) x1 or x1+
  2. §14.8(ii) x1+
  3. §14.8(iii) x

§14.8(i) x1 or x1+

As x1,

14.8.1 𝖯νμ(x) 1Γ(1μ)(21x)μ/2,
μ1,2,3,,
14.8.2 𝖯νm(x) (1)m(νm+1)2mm!(1x2)m/2,
m=1,2,3,, νm1,m2,,m,
14.8.3 𝖰ν(x) =12ln(21x)γψ(ν+1)+O((1x)ln(1x)),
ν1,2,3,,

where γ is Euler’s constant (§5.2(ii)). In the next three relations μ>0.

14.8.4 𝖰νμ(x)12cos(μπ)Γ(μ)(21x)μ/2,
μ12,32,52,,
14.8.5 𝖰νμ(x)(1)μ+(1/2)πΓ(ν+μ+1)2Γ(μ+1)Γ(νμ+1)(1x2)μ/2,
μ=12,32,52,, ν±μ1,2,3,,
14.8.6 𝖰νμ(x)Γ(μ)Γ(νμ+1)2Γ(ν+μ+1)(21x)μ/2,
ν±μ1,2,3,.

The behavior of 𝖯νμ(x) and 𝖰νμ(x) as x1+ follows from the above results and the connection formulas (14.9.8) and (14.9.10).

§14.8(ii) x1+

14.8.7 Pνμ(x) 1Γ(1μ)(2x1)μ/2,
μ1,2,3,,
14.8.8 Pνm(x) Γ(ν+m+1)m!Γ(νm+1)(x12)m/2,
m=1,2,3,, ν±m1,2,3,,
14.8.9 𝑸ν(x) =ln(x1)2Γ(ν+1)+12ln2γψ(ν+1)Γ(ν+1)+O((x1)ln(x1)),
ν1,2,3,,
14.8.10 𝑸n(x)(1)n+1(n1)!,
n=1,2,3,,
14.8.11 𝑸νμ(x)Γ(μ)2Γ(ν+μ+1)(2x1)μ/2,
μ>0, ν+μ1,2,3,.

§14.8(iii) x

14.8.12 Pνμ(x) Γ(ν+12)π1/2Γ(νμ+1)(2x)ν,
ν>12, μν1,2,3,,
14.8.13 Pνμ(x) Γ(ν12)π1/2Γ(μν)(2x)ν+1,
ν<12, ν+μ0,1,2,,
14.8.14 P1/2μ(x) 1Γ(12μ)(2πx)1/2lnx,
μ12,32,52,,
14.8.15 𝑸νμ(x)π1/2Γ(ν+32)(2x)ν+1,
ν32,52,72,,
14.8.16 𝑸n(1/2)μ(x)π1/2Γ(μ+n+12)n!Γ(μn+12)(2x)n+(1/2),
n=1,2,3,, μn+120,1,2,.