Digital Library of Mathematical Functions
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14 Legendre and Related FunctionsComplex Arguments

§14.21 Definitions and Basic Properties


§14.21(i) Associated Legendre Equation

14.21.1 (1-z2)2wz2-2zwz+(ν(ν+1)-μ21-z2)w=0.

Standard solutions: the associated Legendre functions Pνμ(z), Pν-μ(z), Qνμ(z), and Q-ν-1μ(z). Pν±μ(z) and Qνμ(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 Pν±μ(z), Qνμ(z), and Qνμ(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 Pν±μ(z) and Qνμ(z) are real when ν, μ and z(1,).

§14.21(ii) Numerically Satisfactory Solutions

When ν-12 and μ0, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane |phz|12π is given by Pν-μ(z) and Qνμ(z).

§14.21(iii) Properties

Many of the properties stated in preceding sections extend immediately from the x-interval (1,) to the cut z-plane \(-,1]. This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). The generating function expansions (14.7.19) (with P replaced by P) and (14.7.22) apply when |h|<min|z±(z2-1)1/2|; (14.7.21) (with P replaced by P) applies when |h|>max|z±(z2-1)1/2|.