§14.2 Differential Equations

14.2.1		$$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}+\nu (\nu +1)w=0.$$
ⓘ Symbols: $\frac{df}{dx}$: 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 Ch.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 $\nu \in ℝ$ and $x\in (-1,1)$, and $P_{\nu }\left(x\right)$ and $Q_{\nu }\left(x\right)$ are real when $\nu \in ℝ$ and $x\in (1,\mathrm{\infty })$.

14.2.2		$$\left(1-x^2\right)\frac{d^{2}w}{{dx}^2}-2x\frac{dw}{dx}+\left(\nu (\nu +1)-\frac{{\mu }^2}{1-x^2}\right)w=0.$$
ⓘ Symbols: $\frac{df}{dx}$: 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), §14.2(iii), §14.20(i), §14.3(ii), §14.3(i), §14.3(ii), 2nd item, §30.2(iii), Erratum (V1.0.22) for Subsection 14.2(iii) Permalink: http://dlmf.nist.gov/14.2.E2 Encodings: TeX, pMML, png See also: Annotations for §14.2(ii), §14.2 and Ch.14

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

(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 ${\mathsf{P}}_{\nu }^0\left(x\right)={\mathsf{P}}_{\nu }\left(x\right)$, ${\mathsf{Q}}_{\nu }^0\left(x\right)={\mathsf{Q}}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${\mathit{Q}}_{\nu }^0\left(x\right)={\mathit{Q}}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$.

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

Unless stated otherwise in §§14.2–14.20 it is assumed that the arguments of the functions ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$ and ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(-1,1)$, and the arguments of the functions $P_{\nu }^{\mu }\left(x\right)$, $Q_{\nu }^{\mu }\left(x\right)$, and ${\mathit{Q}}_{\nu }^{\mu }\left(x\right)$ lie in the interval $(1,\mathrm{\infty })$. For extensions to complex arguments see §§14.21–14.28.

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

When $\mu -\nu \ne 0,-1,-2,\mathrm{\dots }$, and $\mu +\nu \ne -1,-2,-3,\mathrm{\dots }$, ${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)$ and ${\mathsf{P}}_{\nu }^{-\mu }\left(-x\right)$ are linearly independent, and when $\mathrm{\Re }\mu \ge 0$ 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 . When $\mu -\nu =0,-1,-2,\mathrm{\dots }$, or $\mu +\nu =-1,-2,-3,\mathrm{\dots }$, ${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)$ and ${\mathsf{P}}_{\nu }^{-\mu }\left(-x\right)$ are linearly dependent, and in these cases either may be paired with almost any linearly independent solution to form a numerically satisfactory pair.

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

14.2.3		$$𝒲\left\{{\mathsf{P}}_{\nu }^{-\mu }\left(x\right),{\mathsf{P}}_{\nu }^{-\mu }\left(-x\right)\right\}=\frac{2}{\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +\mu +1\right)\left(1-x^2\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, $𝒲$: Wronskian, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.20(i) Permalink: http://dlmf.nist.gov/14.2.E3 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.4		$$𝒲\left\{{\mathsf{P}}_{\nu }^{\mu }\left(x\right),{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\right\}=\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(1-x^2\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, $𝒲$: Wronskian, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.2(iv) Permalink: http://dlmf.nist.gov/14.2.E4 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.5		$${\mathsf{P}}_{\nu +1}^{\mu }\left(x\right){\mathsf{Q}}_{\nu }^{\mu }\left(x\right)-{\mathsf{P}}_{\nu }^{\mu }\left(x\right){\mathsf{Q}}_{\nu +1}^{\mu }\left(x\right)=\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.2(iv) Permalink: http://dlmf.nist.gov/14.2.E5 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.6		$𝒲\left\{{\mathsf{P}}_{\nu }^{-\mu }\left(x\right),{\mathsf{Q}}_{\nu }^{\mu }\left(x\right)\right\}$	$={\displaystyle \frac{\cos \left(\mu \pi \right)}{1-x^2}},$
ⓘ Symbols: : Ferrers function of the first kind, : Ferrers function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.2.E6 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14
14.2.7		$𝒲\left\{P_{\nu }^{-\mu }\left(x\right),P_{\nu }^{\mu }\left(x\right)\right\}$	$=𝒲\left\{{\mathsf{P}}_{\nu }^{-\mu }\left(x\right),{\mathsf{P}}_{\nu }^{\mu }\left(x\right)\right\}={\displaystyle \frac{2\sin \left(\mu \pi \right)}{\pi \left(1-x^2\right)}},$
ⓘ Symbols: : Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: (14.2.7), §14.21(iii), Erratum (V1.0.9) for Equation (14.2.7) 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 Ch.14

14.2.8		$$𝒲\left\{P_{\nu }^{-\mu }\left(x\right),{\mathit{Q}}_{\nu }^{\mu }\left(x\right)\right\}=-\frac{1}{\mathrm{\Gamma }\left(\nu +\mu +1\right)\left(x^2-1\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $𝒲$: Wronskian, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.2.E8 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.9		$$𝒲\left\{{\mathit{Q}}_{\nu }^{\mu }\left(x\right),{\mathit{Q}}_{-\nu -1}^{\mu }\left(x\right)\right\}=\frac{\cos \left(\nu \pi \right)}{x^2-1},$$
ⓘ Symbols: $𝒲$: Wronskian, ${\mathit{Q}}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $x$: real variable, $\mu$: general order and $\nu$: general degree Permalink: http://dlmf.nist.gov/14.2.E9 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.10		$$𝒲\left\{P_{\nu }^{\mu }\left(x\right),Q_{\nu }^{\mu }\left(x\right)\right\}=-{\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)\left(x^2-1\right)},$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree A&S Ref: 8.1.8 (modified) Permalink: http://dlmf.nist.gov/14.2.E10 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14

14.2.11		$$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree Referenced by: §14.21(iii) Permalink: http://dlmf.nist.gov/14.2.E11 Encodings: TeX, pMML, png See also: Annotations for §14.2(iv), §14.2 and Ch.14