# §14.2 Differential Equations

## §14.2(i) Legendre’s Equation

 14.2.1 $\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{% \mathrm{d}w}{\mathrm{d}x}+\nu(\nu+1)w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $x$: real variable and $\nu$: general degree Referenced by: §14.2(ii) Permalink: http://dlmf.nist.gov/14.2.E1 Encodings: TeX, pMML, png See also: Annotations for 14.2(i), 14.2 and 14

Standard solutions: $\mathsf{P}_{\nu}\left(\pm x\right)$, $\mathsf{Q}_{\nu}\left(\pm x\right)$, $\mathsf{Q}_{-\nu-1}\left(\pm x\right)$, $P_{\nu}\left(\pm x\right)$, $Q_{\nu}\left(\pm x\right)$, $Q_{-\nu-1}\left(\pm x\right)$. $\mathsf{P}_{\nu}\left(x\right)$ and $\mathsf{Q}_{\nu}\left(x\right)$ are real when $\nu\in\mathbb{R}$ and $x\in(-1,1)$, and $P_{\nu}\left(x\right)$ and $Q_{\nu}\left(x\right)$ are real when $\nu\in\mathbb{R}$ and $x\in(1,\infty)$.

## §14.2(ii) Associated Legendre Equation

 14.2.2 $\left(1-x^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}x}^{2}}-2x\frac{% \mathrm{d}w}{\mathrm{d}x}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-x^{2}}\right)w=0.$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.1.1 Referenced by: §14.19(i), §14.2(ii), §14.2(iii), §14.2(iii), §14.2(iii), §14.20(i), §14.3(ii), §14.3(i), §14.32, §30.2(iii) Permalink: http://dlmf.nist.gov/14.2.E2 Encodings: TeX, pMML, png See also: Annotations for 14.2(ii), 14.2 and 14

Standard solutions: $\mathsf{P}^{\mu}_{\nu}\left(\pm x\right)$, $\mathsf{P}^{-\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(\pm x\right)$, $\mathsf{Q}^{\mu}_{-\nu-1}\left(\pm x\right)$, $P^{\mu}_{\nu}\left(\pm x\right)$, $P^{-\mu}_{\nu}\left(\pm x\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(\pm x\right)$, $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(\pm x\right)$.

(14.2.2) reduces to (14.2.1) when $\mu=0$. Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$.

$\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ are real when $\nu$, $\mu$, and $\tau\in\mathbb{R}$, and $x\in(-1,1)$; $P^{\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ are real when $\nu$ and $\mu\in\mathbb{R}$, and $x\in(1,\infty)$.

Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$ and $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P^{\mu}_{\nu}\left(x\right)$, $Q^{\mu}_{\nu}\left(x\right)$, and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ lie in the interval $(1,\infty)$. For extensions to complex arguments see §§14.2114.28.

## §14.2(iii) Numerically Satisfactory Solutions

Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\infty$, with exponent pairs $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, and $\left\{-\nu-1,\nu\right\}$, respectively; compare §2.7(i).

When $\mu-\nu\neq 0,-1,-2,\dots$, and $\mu+\nu\neq-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly independent, and when $\Re\mu\geq 0$ they are recessive at $x=1$ and $x=-1$, respectively. Hence they comprise a numerically satisfactory pair of solutions (§2.7(iv)) of (14.2.2) in the interval $-1. When $\mu-\nu=0,-1,-2,\dots$, or $\mu+\nu=-1,-2,-3,\dots$, $\mathsf{P}^{-\mu}_{\nu}\left(x\right)$ and $\mathsf{P}^{-\mu}_{\nu}\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

When $\Re\mu\geq 0$ and $\Re\nu\geq-\frac{1}{2}$, $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ are linearly independent, and recessive at $x=1$ and $x=\infty$, respectively. Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval $1. With the same conditions, $P^{-\mu}_{\nu}\left(-x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval $-\infty.

## §14.2(iv) Wronskians and Cross-Products

 14.2.3 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{-\mu}_{\nu% }\left(-x\right)\right\}=\frac{2}{\Gamma\left(\mu-\nu\right)\Gamma\left(\nu+% \mu+1\right)\left(1-x^{2}\right)},$
 14.2.4 $\mathscr{W}\left\{\mathsf{P}^{\mu}_{\nu}\left(x\right),\mathsf{Q}^{\mu}_{\nu}% \left(x\right)\right\}=\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu% +1\right)\left(1-x^{2}\right)},$
 14.2.5 $\mathsf{P}^{\mu}_{\nu+1}\left(x\right)\mathsf{Q}^{\mu}_{\nu}\left(x\right)-% \mathsf{P}^{\mu}_{\nu}\left(x\right)\mathsf{Q}^{\mu}_{\nu+1}\left(x\right)=% \frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+2\right)},$
 14.2.6 $\displaystyle\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{Q% }^{\mu}_{\nu}\left(x\right)\right\}$ $\displaystyle=\frac{\cos\left(\mu\pi\right)}{1-x^{2}},$ 14.2.7 $\displaystyle\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{\mu}_{\nu}\left% (x\right)\right\}$ $\displaystyle=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{% P}^{\mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi\right)}{\pi\left(% 1-x^{2}\right)},$ ⓘ Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$: associated Legendre function of the first kind, $\mathscr{W}$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: 14.2.7, §14.21(iii), Other Changes Permalink: http://dlmf.nist.gov/14.2.E7 Encodings: TeX, pMML, png Addition (effective with 1.0.9): (14.2.7) has been expanded to provide the Wronskian for Ferrers functions as well as for associated Legendre functions. Suggested 2014-05-22 by Howard Cohl See also: Annotations for 14.2(iv), 14.2 and 14
 14.2.8 $\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),\boldsymbol{Q}^{\mu}_{\nu}\left% (x\right)\right\}=-\frac{1}{\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)},$
 14.2.9 $\mathscr{W}\left\{\boldsymbol{Q}^{\mu}_{\nu}\left(x\right),\boldsymbol{Q}^{\mu% }_{-\nu-1}\left(x\right)\right\}=\frac{\cos\left(\nu\pi\right)}{x^{2}-1},$
 14.2.10 $\mathscr{W}\left\{P^{\mu}_{\nu}\left(x\right),Q^{\mu}_{\nu}\left(x\right)% \right\}=-e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+% 1\right)\left(x^{2}-1\right)},$
 14.2.11 $P^{\mu}_{\nu+1}\left(x\right)Q^{\mu}_{\nu}\left(x\right)-P^{\mu}_{\nu}\left(x% \right)Q^{\mu}_{\nu+1}\left(x\right)=e^{\mu\pi i}\frac{\Gamma\left(\nu+\mu+1% \right)}{\Gamma\left(\nu-\mu+2\right)}.$