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14 Legendre and Related FunctionsComplex Arguments

§14.23 Values on the Cut

When -1<x<1,

14.23.1 Pνμ(x±i0)=eμπi/2Pνμ(x),
14.23.2 Qνμ(x±i0)=e±μπi/2Γ(ν+μ+1)(Qνμ(x)12πiPνμ(x)).

In terms of the hypergeometric function F14.3(i))

14.23.3 Qνμ(x±i0)=eνπi/2π3/2(1-x2)μ/22ν+1×(xF(12μ-12ν+12,12ν+12μ+1;32;x2)Γ(12ν-12μ+12)Γ(12ν+12μ+12)iF(12μ-12ν,12ν+12μ+12;12;x2)Γ(12ν-12μ+1)Γ(12ν+12μ+1).)

Conversely,

14.23.4 Pνμ(x) =e±μπi/2Pνμ(x±i0),
14.23.5 Qνμ(x) =12Γ(ν+μ+1)(e-μπi/2Qνμ(x+i0)+eμπi/2Qνμ(x-i0)),

or equivalently,

14.23.6 Qνμ(x)=eμπi/2Γ(ν+μ+1)Qνμ(x±i0)±12πie±μπi/2Pνμ(x±i0).

If cuts are introduced along the intervals (-,-1] and [1,), then (14.23.4) and (14.23.6) could be used to extend the definitions of Pνμ(x) and Qνμ(x) to complex x.

The conical function defined by (14.20.2) can be represented similarly by

14.23.7 Q^-12+iτ-μ(x)=12e3μπi/2Q-12+iτ-μ(x-i0)+12e-3μπi/2Q-12-iτ-μ(x+i0).