# Ferrers functions

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## 21—30 of 47 matching pages

##### 21: 29.8 Integral Equations
29.8.2 $\mu w(z_{1})w(z_{2})w(z_{3})=\int_{-2K}^{2K}\mathsf{P}_{\nu}\left(x\right)w(z)% \mathrm{d}z,$
where $\mathsf{P}_{\nu}\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), …
29.8.5 $\mathit{Ec}^{2m}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(K)-w_{2}(-K)}{\left.% \ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}z}\right|_{z=0}}=\int_{-K}^{K}\mathsf{P}_% {\nu}\left(y\right)\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)\mathrm{d}z,$
29.8.7 $\mathit{Ec}^{2m+1}_{\nu}\left(z_{1},k^{2}\right)\frac{w_{2}(K)+w_{2}(-K)}{w_{2% }(0)}=-k^{2}\operatorname{sn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\frac{\mathrm{d}\mathsf{P}_{\nu}\left(y\right)}{\mathrm{d}y}% \mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)\mathrm{d}z,$
29.8.9 $\mathit{Es}^{2m+2}_{\nu}\left(z_{1},k^{2}\right)\frac{\left.\ifrac{\mathrm{d}w% _{2}(z)}{\mathrm{d}z}\right|_{z=K}-\left.\ifrac{\mathrm{d}w_{2}(z)}{\mathrm{d}% z}\right|_{z=-K}}{w_{2}(0)}=-\frac{k^{4}}{k^{\prime}}\operatorname{sn}\left(z_% {1},k\right)\operatorname{cn}\left(z_{1},k\right)\int_{-K}^{K}\operatorname{sn% }\left(z,k\right)\operatorname{cn}\left(z,k\right)\frac{{\mathrm{d}}^{2}% \mathsf{P}_{\nu}\left(y\right)}{{\mathrm{d}y}^{2}}\mathit{Es}^{2m+2}_{\nu}% \left(z,k^{2}\right)\mathrm{d}z.$
##### 22: 16.18 Special Cases
As a corollary, special cases of the ${{}_{1}F_{1}}$ and ${{}_{2}F_{1}}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. …
##### 24: 14.15 Uniform Asymptotic Approximations
###### §14.15(i) Large $\mu$, Fixed $\nu$
For asymptotic expansions and explicit error bounds, see Dunster (2003b).
##### 25: 14.30 Spherical and Spheroidal Harmonics
###### §14.30 Spherical and Spheroidal Harmonics
14.30.2 $Y_{l}^{m}\left(\theta,\phi\right)=\cos\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right)\text{ or }\sin\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right).$
Most mathematical properties of $Y_{{l},{m}}\left(\theta,\phi\right)$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter. …
14.30.9 $\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).$
##### 26: 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)). …
##### 27: 15.9 Relations to Other Functions
###### §15.9(iv) Associated Legendre Functions; FerrersFunctions
Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. …
##### 28: 14.20 Conical (or Mehler) Functions
For $-1 and $\tau>0$, a numerically satisfactory pair of real conical functions is $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ and $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(-x\right)$. …
14.20.4 $\mathscr{W}\left\{\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right% ),\mathsf{P}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(-x\right)\right\}=\frac% {2}{|\Gamma\left(\mu+\frac{1}{2}+\mathrm{i}\tau\right)|^{2}(1-x^{2})}.$
14.20.7 $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)\sim\tfrac{1}{2}% \Gamma\left(\mu\right)\left(\frac{2}{1-x}\right)^{\mu/2},$
14.20.22 $\mathsf{P}^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)=\frac{\beta\exp\left(\mu% \beta\operatorname{arctan}\beta\right)}{\Gamma\left(\mu+1\right)\left(1+\beta^% {2}\right)^{\mu/2}}\frac{e^{-\mu\rho}}{\left(1+\beta^{2}-x^{2}\beta^{2}\right)% ^{1/4}}\left(1+O\left(\frac{1}{\mu}\right)\right),$
##### 30: 30.5 Functions of the Second Kind
30.5.3 $\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$