Digital Library of Mathematical Functions
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14 Legendre and Related FunctionsComplex Arguments

§14.21 Definitions and Basic Properties

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Referenced by:
§14.2(ii)
Permalink:
http://dlmf.nist.gov/14.21
Contents

§14.21(i) Associated Legendre Equation

Standard solutions: the associated Legendre functions \mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right), \mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(z\right), \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right), and \mathop{\boldsymbol{Q}^{{\mu}}_{{-\nu-1}}\/}\nolimits\!\left(z\right). \mathop{P^{{\pm\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) and \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) exist for all values of \nu, \mu, and z, except possibly z=\pm 1 and \infty, which are branch points (or poles) of the functions, in general. When z is complex \mathop{P^{{\pm\mu}}_{{\nu}}\/}\nolimits\!\left(z\right), \mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right), and \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) 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\in(1,\infty), and by continuity elsewhere in the z-plane with a cut along the interval (-\infty,1]; compare §4.2(i). The principal branches of \mathop{P^{{\pm\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) and \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) are real when \nu, \mu\in\Real and z\in(1,\infty).

§14.21(ii) Numerically Satisfactory Solutions

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Notes:
See Olver (1997b, p. 172).
Keywords:
associated Legendre equation
Permalink:
http://dlmf.nist.gov/14.21.ii

When \realpart{\nu}\geq-\frac{1}{2} and \realpart{\mu}\geq 0, a numerically satisfactory pair of solutions of (14.21.1) in the half-plane |\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi is given by \mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(z\right) and \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right).

§14.21(iii) Properties

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Notes:
See Erdélyi et al. (1953a, Chapter 3) and Olver (1997b, Chapter 5, §§12–14).
Keywords:
associated Legendre functions
Referenced by:
§15.9(iv)
Permalink:
http://dlmf.nist.gov/14.21.iii

Many of the properties stated in preceding sections extend immediately from the x-interval (1,\infty) to the cut z-plane \Complex\backslash(-\infty,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 \mathop{\mathsf{P}\/}\nolimits replaced by \mathop{P\/}\nolimits) and (14.7.22) apply when |h|<\min\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|; (14.7.21) (with \mathop{\mathsf{P}\/}\nolimits replaced by \mathop{P\/}\nolimits) applies when |h|>\max\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|.

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