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14 Legendre and Related FunctionsComplex Arguments

§14.24 Analytic Continuation

Let s be an arbitrary integer, and Pνμ(zesπi) and 𝑸νμ(zesπi) denote the branches obtained from the principal branches by making 12s circuits, in the positive sense, of the ellipse having ±1 as foci and passing through z. Then

14.24.1 Pνμ(zesπi)=esνπiPνμ(z)+2isin((ν+12)sπ)esπi/2cos(νπ)Γ(μν)𝑸νμ(z),
14.24.2 𝑸νμ(zesπi)=(1)sesνπi𝑸νμ(z),

the limiting value being taken in (14.24.1) when 2ν is an odd integer.

Next, let Pν,sμ(z) and 𝑸ν,sμ(z) denote the branches obtained from the principal branches by encircling the branch point 1 (but not the branch point 1) s times in the positive sense. Then

14.24.3 Pν,sμ(z) =esμπiPνμ(z),
14.24.4 𝑸ν,sμ(z) =esμπi𝑸νμ(z)πisin(sμπ)sin(μπ)Γ(νμ+1)Pνμ(z),

the limiting value being taken in (14.24.4) when μ.

For fixed z, other than ±1 or , each branch of Pνμ(z) and 𝑸νμ(z) is an entire function of each parameter ν and μ.

The behavior of Pνμ(z) and 𝑸νμ(z) as z1 from the left on the upper or lower side of the cut from to 1 can be deduced from (14.8.7)–(14.8.11), combined with (14.24.1) and (14.24.2) with s=±1.