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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(1x2)μ/22ν+1×(x𝐅(12μ12ν+12,12ν+12μ+1;32;x2)Γ(12ν12μ+12)Γ(12ν+12μ+12)i𝐅(12μ12ν,12ν+12μ+12;12;x2)Γ(12ν12μ+1)Γ(12ν+12μ+1)).


14.23.4 𝖯νμ(x) =e±μπi/2Pνμ(x±i0),
14.23.5 𝖰νμ(x) =12Γ(ν+μ+1)(eμπi/2𝑸νμ(x+i0)+eμπi/2𝑸νμ(xi0)),

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+iτμ(x)=12e3μπi/2Q12+iτμ(xi0)+12e3μπi/2Q12iτμ(x+i0).