# Legendre equation

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##### 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).
##### 4: 18.39 Physical Applications
A second example is provided by the three-dimensional time-independent Schrödinger equation …The eigenfunctions of one of the separated ordinary differential equations are Legendre polynomials. …
##### 6: 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)). …
##### 8: 30.2 Differential Equations
If $\gamma=0$, Equation (30.2.1) is the associated Legendre differential equation; see (14.2.2). …
##### 9: 14.3 Definitions and Hypergeometric Representations
14.3.8 $P^{m}_{\nu}\left(x\right)=\frac{\Gamma\left(\nu+m+1\right)}{2^{m}\Gamma\left(% \nu-m+1\right)}\left(x^{2}-1\right)^{m/2}\mathbf{F}\left(\nu+m+1,m-\nu;m+1;% \tfrac{1}{2}-\tfrac{1}{2}x\right).$
##### 10: Howard S. Cohl
Cohl has published papers in orthogonal polynomials and special functions, and is particularly interested in fundamental solutions of linear partial differential equations on Riemannian manifolds, associated Legendre functions, generalized and basic hypergeometric functions, eigenfunction expansions of fundamental solutions in separable coordinate systems for linear partial differential equations, orthogonal polynomial generating function and generalized expansions, and $q$-series. …