§14.21 Definitions and Basic Properties

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

§14.21(i) Associated Legendre Equation
§14.21(ii) Numerically Satisfactory Solutions
§14.21(iii) Properties

§14.21(i) Associated Legendre Equation

Notes:: See Olver (1997b, p. 169).
Keywords:: Olver’s associated Legendre function, associated Legendre equation, associated Legendre functions
Referenced by:: §10.43(iv), §10.54, §14.25, §15.9(iv)
Permalink:: http://dlmf.nist.gov/14.21.i

14.21.1 $\left(1-z^{2}\right)\frac{{d}^{2}w}{{dz}^{2}}-2z\frac{dw}{dz}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.1.1
Referenced by:: §14.21(ii), §14.29, §14.32
Permalink:: http://dlmf.nist.gov/14.21.E1
Encodings:: TeX, pMML, png

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

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

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 $|h|>\max\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|$ .

§14.21 Definitions and Basic Properties

Contents

§14.21(i) Associated Legendre Equation

§14.21(ii) Numerically Satisfactory Solutions

§14.21(iii) Properties