§14.2 Differential Equations

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

§14.2(i) Legendre’s Equation
§14.2(ii) Associated Legendre Equation
§14.2(iii) Numerically Satisfactory Solutions
§14.2(iv) Wronskians and Cross-Products

§14.2(i) Legendre’s Equation

Notes:: See Olver (1997b, p. 169).
Keywords:: Legendre’s equation
Permalink:: http://dlmf.nist.gov/14.2.i

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

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

Notes:: See Olver (1997b, p. 169).
Keywords:: Ferrers functions, associated Legendre equation, associated Legendre functions
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/14.2.ii
Errata (effective with 1.0.5):: Originally it was stated, incorrectly, that $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is real when $\nu,\mu\in\Real$ and $x\in(1,\infty)$ . This statement is true only for $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ .
Reported 2012-07-18 by Hans Volkmer and Howard Cohl

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), §14.20(i), ¶ ‣ §14.3(ii), §14.3(i), §14.32, §30.2(iii)
Permalink:: http://dlmf.nist.gov/14.2.E2
Encodings:: TeX, pMML, png

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 §§14.2–14.20 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 §§14.21–14.28.

§14.2(iii) Numerically Satisfactory Solutions

Notes:: See Olver (1997b, p. 172) for the pair $\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ . The result for $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right)$ follows from the fact that when $\realpart{\mu}\geq 0$ , $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ is recessive as $x\to 1-$ and $\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right)$ is recessive as $x\to-1+$ ; see §14.8(i).
Keywords:: associated Legendre equation, singularities
Permalink:: http://dlmf.nist.gov/14.2.iii

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 $-1<x<1$ . 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 $1<x<\infty$ . 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 $-\infty<x<-1$ .

§14.2(iv) Wronskians and Cross-Products

Notes:: For (14.2.4) and (14.2.5) see Olver (1997b, p. 172). The other results may be derived in a similar manner, or by application of the connection formulas in §14.9.
Keywords:: Ferrers functions, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.2.iv

14.2.3 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(-x\right)\right\}=\frac{2}{\mathop{\Gamma\/}\nolimits\!\left(\mu-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(1-x^{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathscr{W}\/}\nolimits$ : 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

14.2.4 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right\}=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)\left(1-x^{2}\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\mathscr{W}\/}\nolimits$ : 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

14.2.5 $\mathop{\mathsf{P}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{\mathsf{Q}^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+2\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : 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

14.2.6 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{P}^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right\}=\frac{\mathop{\cos\/}\nolimits\!\left(\mu\pi\right)}{1-x^{2}},$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\cos\/}\nolimits z$ : cosine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.2.E6
Encodings:: TeX, pMML, png

14.2.7 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right\}=-\frac{2\mathop{\sin\/}\nolimits\!\left(\mu\pi\right)}{\pi\left(x^{2}-1\right)},$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.21(iii)
Permalink:: http://dlmf.nist.gov/14.2.E7
Encodings:: TeX, pMML, png

14.2.8 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{{-\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right\}=-\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(x^{2}-1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : 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

14.2.9 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{\boldsymbol{Q}^{{\mu}}_{{-\nu-1}}\/}\nolimits\!\left(x\right)\right\}=\frac{\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)}{x^{2}-1},$

Symbols:: $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, $\mathop{\cos\/}\nolimits z$ : cosine function, $x$ : real variable, $\mu$ : general order and $\nu$ : general degree
Permalink:: http://dlmf.nist.gov/14.2.E9
Encodings:: TeX, pMML, png

14.2.10 $\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right\}=-e^{{\mu\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+1\right)\left(x^{2}-1\right)},$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $\mathop{\mathscr{W}\/}\nolimits$ : Wronskian, $e$ : base of exponential function, $x$ : 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

14.2.11 $\mathop{P^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)-\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\mathop{Q^{{\mu}}_{{\nu+1}}\/}\nolimits\!\left(x\right)=e^{{\mu\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-\mu+2\right)}.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, $e$ : base of exponential function, $x$ : 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

§14.2 Differential Equations

Contents

§14.2(i) Legendre’s Equation

§14.2(ii) Associated Legendre Equation

§14.2(iii) Numerically Satisfactory Solutions

§14.2(iv) Wronskians and Cross-Products