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14 Legendre and Related FunctionsComplex Arguments

§14.25 Integral Representations

The principal values of Pν-μ(z) and Qνμ(z)14.21(i)) are given by

14.25.1 Pν-μ(z)=(z2-1)μ/22νΓ(μ-ν)Γ(ν+1)0(sinht)2ν+1(z+cosht)ν+μ+1dt,
μ>ν>-1,
14.25.2 Qνμ(z)=π1/2(z2-1)μ/22μΓ(μ+12)Γ(ν-μ+1)0(sinht)2μ(z+(z2-1)1/2cosht)ν+μ+1dt,
(ν+1)>μ>-12,

where the multivalued functions have their principal values when 1<z< and are continuous in \(-,1].

For corresponding contour integrals, with less restrictions on μ and ν, see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).