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14 Legendre and Related FunctionsComplex Arguments

§14.27 Zeros

Pνμ(x±i0) (either side of the cut) has exactly one zero in the interval (-,-1) if either of the following sets of conditions holds:

  • (a)

    μ<0, μ, ν, and sin((μ-ν)π) and sin(μπ) have opposite signs.

  • (b)

    μ,ν, μ+ν<0, and ν is odd.

For all other values of the parameters Pνμ(x±i0) has no zeros in the interval (-,-1).

For complex zeros of Pνμ(z) see Hobson (1931, §§233, 234, and 238).