# §14.2(i) Legendre’s Equation

 14.2.1 $\left(1-x^{2}\right)\frac{{d}^{2}w}{{dx}^{2}}-2x\frac{dw}{dx}+\nu(\nu+1)w=0.$

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

# §14.2(ii) Associated Legendre Equation

 14.2.2 $\left(1-x^{2}\right)\frac{{d}^{2}w}{{dx}^{2}}-2x\frac{dw}{dx}+\left(\nu(\nu+1)% -\frac{\mu^{2}}{1-x^{2}}\right)w=0.$

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

(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 $\mathop{\mathsf{P}^{0}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{\mathsf{P}_{% \nu}\/}\nolimits\!\left(x\right)$, $\mathop{\mathsf{Q}^{0}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{\mathsf{Q}_{% \nu}\/}\nolimits\!\left(x\right)$, $\mathop{P^{0}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{P_{\nu}\/}\nolimits\!% \left(x\right)$, $\mathop{Q^{0}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{Q_{\nu}\/}\nolimits\!% \left(x\right)$, $\mathop{\boldsymbol{Q}^{0}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{% \boldsymbol{Q}_{\nu}\/}\nolimits\!\left(x\right)=\mathop{Q_{\nu}\/}\nolimits\!% \left(x\right)/\mathop{\Gamma\/}\nolimits\!\left(\nu+1\right)$.

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

Unless stated otherwise in §§14.214.20 it is assumed that the arguments of the functions $\mathop{\mathsf{P}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, $\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$, and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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 $\mu+\nu\neq-1,-2,-3,\dots$, $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(-x\right)$ are linearly independent, and when $\realpart{\mu}\geq 0$ 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 $\mu+\nu=-1,-2,-3,\dots$, $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

When $\realpart{\mu}\geq 0$ and $\realpart{\nu}\geq-\frac{1}{2}$, $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ 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, $\mathop{P^{-\mu}_{\nu}\/}\nolimits\!\left(-x\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(-x\right)$ comprise a numerically satisfactory pair of solutions in the interval $-\infty.

# §14.2(iv) Wronskians and Cross-Products

 14.2.3 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{-\mu}_{\nu}\/}% \nolimits\!\left(x\right),\mathop{\mathsf{P}^{-\mu}_{\nu}\/}\nolimits\!\left(-% x\right)\right\}=\frac{2}{\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)% \mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(1-x^{2}\right)},$
 14.2.4 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right),\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x% \right)\right\}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)\left(1-x^{2}\right)},$
 14.2.5 $\mathop{\mathsf{P}^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}% ^{\mu}_{\nu}\/}\nolimits\!\left(x\right)-\mathop{\mathsf{P}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)\mathop{\mathsf{Q}^{\mu}_{\nu+1}\/}\nolimits\!\left(x% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{% \Gamma\/}\nolimits\!\left(\nu-\mu+2\right)},$
 14.2.6 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{-\mu}_{% \nu}\/}\nolimits\!\left(x\right),\mathop{\mathsf{Q}^{\mu}_{\nu}\/}\nolimits\!% \left(x\right)\right\}$ $\displaystyle=\frac{\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)}{1-x^{2}},$ 14.2.7 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{-\mu}_{\nu}\/}% \nolimits\!\left(x\right),\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\right\}$ $\displaystyle=-\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi\left(% x^{2}-1\right)},$
 14.2.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{-\mu}_{\nu}\/}\nolimits\!% \left(x\right),\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)% \right\}=-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(x^{% 2}-1\right)},$
 14.2.9 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right),\mathop{\boldsymbol{Q}^{\mu}_{-\nu-1}\/}\nolimits\!% \left(x\right)\right\}=\frac{\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)}{x^% {2}-1},$
 14.2.10 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left% (x\right),\mathop{Q^{\mu}_{\nu}\/}\nolimits\!\left(x\right)\right\}=-e^{\mu\pi i% }\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}% \nolimits\!\left(\nu-\mu+1\right)\left(x^{2}-1\right)},$
 14.2.11 $\mathop{P^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)\mathop{Q^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)-\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)% \mathop{Q^{\mu}_{\nu+1}\/}\nolimits\!\left(x\right)=e^{\mu\pi i}\frac{\mathop{% \Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(% \nu-\mu+2\right)}.$