# associated Legendre functions

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##### 1: 14.1 Special Notation
###### §14.1 Special Notation
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 2: 14.21 Definitions and Basic Properties
14.21.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$
Standard solutions: the associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $P^{-\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$, and $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(z\right)$. $P^{\pm\mu}_{\nu}\left(z\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$ exist for all values of $\nu$, $\mu$, and $z$, except possibly $z=\pm 1$ and $\infty$, which are branch points (or poles) of the functions, in general. … …
##### 3: 14.31 Other Applications
###### §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)). …
##### 5: 14.29 Generalizations
are called Generalized Associated Legendre Functions. … …
##### 6: 14.22 Graphics
###### §14.22 Graphics Figure 14.22.4: 𝑸 0 0 ⁡ ( x + i ⁢ y ) , − 5 ≤ x ≤ 5 , − 5 ≤ y ≤ 5 . … Magnify 3D Help
##### 7: 14.25 Integral Representations
###### §14.25 Integral Representations
14.25.1 $P^{-\mu}_{\nu}\left(z\right)=\frac{\left(z^{2}-1\right)^{\mu/2}}{2^{\nu}\Gamma% \left(\mu-\nu\right)\Gamma\left(\nu+1\right)}\int_{0}^{\infty}\frac{(\sinh t)^% {2\nu+1}}{(z+\cosh t)^{\nu+\mu+1}}\,\mathrm{d}t,$ $\Re\mu>\Re\nu>-1$,
14.25.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)=\frac{\pi^{1/2}\left(z^{2}-1\right)^{% \mu/2}}{2^{\mu}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}% \*\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{\left(z+(z^{2}-1)^{1/2}\cosh t% \right)^{\nu+\mu+1}}\,\mathrm{d}t,$ $\Re\left(\nu+1\right)>\Re\mu>-\tfrac{1}{2}$,
##### 9: 14.28 Sums
###### §14.28 Sums
14.28.1 $P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)% ^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum% _{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1% \right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right),$