# Legendre equation

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##### 1: 14.21 Definitions and Basic Properties
###### §14.21(i) Associated LegendreEquation
14.21.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$
##### 2: 14.2 Differential Equations
###### §14.2(ii) Associated LegendreEquation
Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations $\mathsf{P}^{0}_{\nu}\left(x\right)=\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}^{0}_{\nu}\left(x\right)=\mathsf{Q}_{\nu}\left(x\right)$, $P^{0}_{\nu}\left(x\right)=P_{\nu}\left(x\right)$, $Q^{0}_{\nu}\left(x\right)=Q_{\nu}\left(x\right)$, $\boldsymbol{Q}^{0}_{\nu}\left(x\right)=\boldsymbol{Q}_{\nu}\left(x\right)=Q_{% \nu}\left(x\right)/\Gamma\left(\nu+1\right)$. …
###### §14.2(iii) Numerically Satisfactory Solutions
14.2.7 $\mathscr{W}\left\{P^{-\mu}_{\nu}\left(x\right),P^{\mu}_{\nu}\left(x\right)% \right\}=\mathscr{W}\left\{\mathsf{P}^{-\mu}_{\nu}\left(x\right),\mathsf{P}^{% \mu}_{\nu}\left(x\right)\right\}=\frac{2\sin\left(\mu\pi\right)}{\pi\left(1-x^% {2}\right)},$
##### 3: 14.29 Generalizations
For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).
##### 5: 14.31 Other Applications
The conical functions $\mathsf{P}^{m}_{-\frac{1}{2}+i\tau}\left(x\right)$ appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
###### §14.31(iii) Miscellaneous
Many additional physical applications of Legendre polynomials and associated Legendre functions include solution of the Helmholtz equation, as well as the Laplace equation, in spherical coordinates (Temme (1996b)), quantum mechanics (Edmonds (1974)), and high-frequency scattering by a sphere (Nussenzveig (1965)). …
##### 6: 19.4 Derivatives and Differential Equations
19.4.3 $\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{% \mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-% E\left(k\right)}{k^{2}{k^{\prime}}^{2}},$
19.4.6 $\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F% \left(\phi,k\right)}{k},$
###### §19.4(ii) Differential Equations
19.4.8 $(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k% \sin\phi\cos\phi}{(1-k^{2}{\sin}^{2}\phi)^{3/2}},$
19.4.9 $(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac% {k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin}^{2}\phi}}.$
##### 7: 14.3 Definitions and Hypergeometric Representations
14.3.6 $P^{\mu}_{\nu}\left(x\right)=\left(\frac{x+1}{x-1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
14.3.9 $P^{-\mu}_{\nu}\left(x\right)=\left(\frac{x-1}{x+1}\right)^{\mu/2}\mathbf{F}% \left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
14.3.11 $\mathsf{P}^{\mu}_{\nu}\left(x\right)=\cos\left(\tfrac{1}{2}(\nu+\mu)\pi\right)% w_{1}(\nu,\mu,x)+\sin\left(\tfrac{1}{2}(\nu+\mu)\pi\right)w_{2}(\nu,\mu,x),$
14.3.15 $P^{-\mu}_{\nu}\left(x\right)=2^{-\mu}\left(x^{2}-1\right)^{\mu/2}\mathbf{F}% \left(\mu-\nu,\nu+\mu+1;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right),$
##### 8: 30.2 Differential Equations
If $\gamma=0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …
##### 9: 14.5 Special Values
14.5.30 $\boldsymbol{Q}_{2}\left(x\right)=\frac{3x^{2}-1}{8}\ln\left(\frac{x+1}{x-1}% \right)-\frac{3}{4}x.$
##### 10: 14.12 Integral Representations
14.12.5 $P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(% \mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\,\mathrm{d}t,$ $\Re\mu>0$.
14.12.11 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{\left(x^{2}-1\right)^{m/2}}{2^{n+1}% n!}\int_{-1}^{1}\frac{\left(1-t^{2}\right)^{n}}{(x-t)^{n+m+1}}\,\mathrm{d}t,$
14.12.12 $\boldsymbol{Q}^{m}_{n}\left(x\right)=\frac{1}{(n-m)!}P^{m}_{n}\left(x\right)% \int_{x}^{\infty}\frac{\,\mathrm{d}t}{\left(t^{2}-1\right)\left(\displaystyle P% ^{m}_{n}\left(t\right)\right)^{2}},$ $n\geq m$.
14.12.13 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{2(n!)}\int_{-1}^{1}\frac{P_{n}\left(% t\right)}{x-t}\,\mathrm{d}t.$
14.12.14 $\boldsymbol{Q}_{n}\left(x\right)=\frac{1}{n!}\int_{0}^{\infty}\frac{\,\mathrm{% d}t}{\left(x+(x^{2}-1)^{1/2}\cosh t\right)^{n+1}}.$