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1: 14.27 Zeros
§14.27 Zeros
P ν μ ( x ± i 0 ) (either side of the cut) has exactly one zero in the interval ( , 1 ) if either of the following sets of conditions holds: …For all other values of the parameters P ν μ ( x ± i 0 ) has no zeros in the interval ( , 1 ) . For complex zeros of P ν μ ( z ) see Hobson (1931, §§233, 234, and 238).
2: 14.21 Definitions and Basic Properties
Standard solutions: the associated Legendre functions P ν μ ( z ) , P ν μ ( z ) , 𝑸 ν μ ( z ) , and 𝑸 ν 1 μ ( z ) . P ν ± μ ( z ) and 𝑸 ν μ ( z ) exist for all values of ν , μ , and z , except possibly z = ± 1 and , which are branch points (or poles) of the functions, in general. When z is complex P ν ± μ ( z ) , Q ν μ ( z ) , and 𝑸 ν μ ( z ) are defined by (14.3.6)–(14.3.10) with x replaced by z : the principal branches are obtained by taking the principal values of all the multivalued functions appearing in these representations when z ( 1 , ) , and by continuity elsewhere in the z -plane with a cut along the interval ( , 1 ] ; compare §4.2(i). The principal branches of P ν ± μ ( z ) and 𝑸 ν μ ( z ) are real when ν , μ and z ( 1 , ) . … The generating function expansions (14.7.19) (with 𝖯 replaced by P ) and (14.7.22) apply when | h | < min | z ± ( z 2 1 ) 1 / 2 | ; (14.7.21) (with 𝖯 replaced by P ) applies when | h | > max | z ± ( z 2 1 ) 1 / 2 | .