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14 Legendre and Related FunctionsReal Arguments

§14.8 Behavior at Singularities

Contents

§14.8(i) x1- or x-1+

As x1-,

14.8.1 Pνμ(x) 1Γ(1-μ)(21-x)μ/2,
μ1,2,3,,
14.8.2 Pνm(x) (-1)m(ν-m+1)2mm!(1-x2)m/2,
m=1,2,3,, νm-1,m-2,,-m,
14.8.3 Qν(x) =12ln(21-x)-γ-ψ(ν+1)+O(1-x),
ν-1,-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)(21-x)μ/2,
μ12,32,52,,
14.8.5 Qνμ(x)(-1)μ+(1/2)πΓ(ν+μ+1)2Γ(μ+1)Γ(ν-μ+1)(1-x2)μ/2,
μ=12,32,52,, ν±μ-1,-2,-3,,
14.8.6 Qν-μ(x)Γ(μ)Γ(ν-μ+1)2Γ(ν+μ+1)(21-x)μ/2,
ν±μ-1,-2,-3,.

The behavior of Pνμ(x) and Qνμ(x) as x-1+ 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(1-μ)(2x-1)μ/2,
μ1,2,3,,
14.8.8 Pνm(x) Γ(ν+m+1)m!Γ(ν-m+1)(x-12)m/2,
m=1,2,3,, ν±m-1,-2,-3,,
14.8.9 Qν(x) =-ln(x-1)2Γ(ν+1)+12ln2-γ-ψ(ν+1)Γ(ν+1)+O(x-1),
ν-1,-2,-3,,
14.8.10 Q-n(x)(-1)n+1(n-1)!,
n=1,2,3,,
14.8.11 Qνμ(x)Γ(μ)2Γ(ν+μ+1)(2x-1)μ/2,
μ>0, ν+μ-1,-2,-3,.

§14.8(iii) x

14.8.12 Pνμ(x) Γ(ν+12)π1/2Γ(ν-μ+1)(2x)ν,
ν>-12, μ-ν1,2,3,,
14.8.13 Pνμ(x) Γ(-ν-12)π1/2Γ(-μ-ν)(2x)ν+1,
ν<-12, ν+μ0,1,2,,
14.8.14 P-1/2μ(x) 1Γ(12-μ)(2πx)1/2lnx,
μ12,32,52,,
14.8.15 Qνμ(x)π1/2Γ(ν+32)(2x)ν+1,
ν-32,-52,-72,,
14.8.16 Q-n-(1/2)μ(x)π1/2Γ(μ+n+12)n!Γ(μ-n+12)(2x)n+(1/2),
n=1,2,3,, μ-n+120,-1,-2,.