# 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: 19.2 Definitions
19.2.4 $F\left(\phi,k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{\sqrt{1-k^{2}{% \sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{% 1-k^{2}t^{2}}},$
19.2.5 $E\left(\phi,k\right)=\int_{0}^{\phi}\sqrt{1-k^{2}{\sin}^{2}\theta}\,\mathrm{d}% \theta\\ =\int_{0}^{\sin\phi}\frac{\sqrt{1-k^{2}t^{2}}}{\sqrt{1-t^{2}}}\,\mathrm{d}t.$
19.2.6 $D\left(\phi,k\right)=\int_{0}^{\phi}\frac{{\sin}^{2}\theta\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}}=\int_{0}^{\sin\phi}\frac{t^{2}\,\mathrm{d}t}{% \sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}}=(F\left(\phi,k\right)-E\left(\phi,k\right))% /k^{2}.$
19.2.7 $\Pi\left(\phi,\alpha^{2},k\right)=\int_{0}^{\phi}\frac{\,\mathrm{d}\theta}{% \sqrt{1-k^{2}{\sin}^{2}\theta}(1-\alpha^{2}{\sin}^{2}\theta)}=\int_{0}^{\sin% \phi}\frac{\,\mathrm{d}t}{\sqrt{1-t^{2}}\sqrt{1-k^{2}t^{2}}(1-\alpha^{2}t^{2})}.$
19.2.16 $\operatorname{el3}\left(x,k_{c},p\right)=\int_{0}^{\operatorname{arctan}x}% \frac{\,\mathrm{d}\theta}{({\cos}^{2}\theta+p{\sin}^{2}\theta)\sqrt{{\cos}^{2}% \theta+k_{c}^{2}{\sin}^{2}\theta}}=\Pi\left(\operatorname{arctan}x,1-p,k\right),$ $x^{2}\neq-1/p$.
##### 3: 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)). … Legendre functions $P_{\nu}\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)).
##### 4: 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 $\boldsymbol{Q}^{\mu}_{-\nu-1}\left(z\right)$. … …
###### §14.21(iii) Properties
The generating function expansions (14.7.19) (with $\mathsf{P}$ replaced by $P$) and (14.7.22) apply when $|h|<\min\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$; (14.7.21) (with $\mathsf{P}$ replaced by $P$) applies when $|h|>\max\left|z\pm\left(z^{2}-1\right)^{1/2}\right|$.
##### 6: 14.28 Sums
###### §14.28 Sums
14.28.1 $P_{\nu}\left(z_{1}z_{2}-\left(z_{1}^{2}-1\right)^{1/2}\left(z_{2}^{2}-1\right)% ^{1/2}\cos\phi\right)=P_{\nu}\left(z_{1}\right)P_{\nu}\left(z_{2}\right)+2\sum% _{m=1}^{\infty}(-1)^{m}\frac{\Gamma\left(\nu-m+1\right)}{\Gamma\left(\nu+m+1% \right)}\*P^{m}_{\nu}\left(z_{1}\right)P^{m}_{\nu}(z_{2})\cos\left(m\phi\right),$
14.28.2 $\sum_{n=0}^{\infty}(2n+1)Q_{n}\left(z_{1}\right)P_{n}\left(z_{2}\right)=\frac{% 1}{z_{1}-z_{2}},$ $z_{1}\in\mathcal{E}_{1}$, $z_{2}\in\mathcal{E}_{2}$,
14.6.7 $P^{-m}_{\nu}\left(x\right)=\left(x^{2}-1\right)^{-m/2}\int_{1}^{x}\!\dots\!% \int_{1}^{x}P_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^{m},$
14.6.8 $Q^{-m}_{\nu}\left(x\right)=(-1)^{m}\left(x^{2}-1\right)^{-m/2}\*\int_{x}^{% \infty}\!\dots\!\int_{x}^{\infty}Q_{\nu}\left(x\right)\left(\!\,\mathrm{d}x% \right)^{m}.$