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14 Legendre and Related FunctionsComplex Arguments

§14.24 Analytic Continuation

Let s be an arbitrary integer, and Pν-μ(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 Pν-μ(zesπi)=esνπiPν-μ(z)+2isin((ν+12)sπ)e-sπi/2cos(νπ)Γ(μ-ν)Qνμ(z),
14.24.2 Qνμ(zesπi)=(-1)se-sνπiQνμ(z),

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

Next, let Pν,s-μ(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,s-μ(z) =esμπiPν-μ(z),
14.24.4 Qν,sμ(z) =e-sμπiQνμ(z)-πisin(sμπ)sin(μπ)Γ(ν-μ+1)Pν-μ(z),

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

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

The behavior of Pν-μ(z) and Qνμ(z) as z-1 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.