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14 Legendre and Related FunctionsComplex Arguments

§14.25 Integral Representations

The principal values of Pνμ(z) and 𝑸νμ(z)14.21(i)) are given by

14.25.1 Pνμ(z)=(z21)μ/22νΓ(μν)Γ(ν+1)0(sinht)2ν+1(z+cosht)ν+μ+1dt,
14.25.2 𝑸νμ(z)=π1/2(z21)μ/22μΓ(μ+12)Γ(νμ+1)0(sinht)2μ(z+(z21)1/2cosht)ν+μ+1dt,

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).