29 Lamé Functions Applications 29.17 Other Solutions 29.19 Physical Applications

§29.18 Mathematical Applications

Referenced by:: ¶ ‣ §1.5(ii)
Permalink:: http://dlmf.nist.gov/29.18

§29.18(i) Sphero-Conal Coordinates
§29.18(ii) Ellipsoidal Coordinates
§29.18(iii) Spherical and Ellipsoidal Harmonics
§29.18(iv) Other Applications

§29.18(i) Sphero-Conal Coordinates

Notes:: See Erdélyi et al. (1955, §15.1.2) and Jansen (1977).
Keywords:: Lamé functions, coordinate systems, sphero-conal coordinates, wave equation
Permalink:: http://dlmf.nist.gov/29.18.i

The wave equation

29.18.1 $\nabla^{2}u+\omega^{2}u=0,$

Symbols:: : solution and $\omega$ : parameter
Referenced by:: §29.18(ii)
Permalink:: http://dlmf.nist.gov/29.18.E1
Encodings:: TeX, pMML, png

when transformed to sphero-conal coordinates $r,\beta,\gamma$ :

29.18.2

$x=kr\mathop{\mathrm{sn}\/}\nolimits\left(\beta,k\right)\mathop{\mathrm{sn}\/}% \nolimits\left(\gamma,k\right),$

$y=i\frac{k}{k^{{\prime}}}r\mathop{\mathrm{cn}\/}\nolimits\left(\beta,k\right)% \mathop{\mathrm{cn}\/}\nolimits\left(\gamma,k\right),$

$z=\frac{1}{k^{{\prime}}}r\mathop{\mathrm{dn}\/}\nolimits\left(\beta,k\right)% \mathop{\mathrm{dn}\/}\nolimits\left(\gamma,k\right),$

with

29.18.3

$r\geq 0,$

$\beta=\!\mathop{K\/}\nolimits\!+i\beta^{{\prime}},$

$0\leq\beta^{{\prime}}\leq 2\!\mathop{{K^{{\prime}}}\/}\nolimits\!,$

$0\leq\gamma\leq 4\!\mathop{K\/}\nolimits\!,$

admits solutions

29.18.4 $u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(\gamma),$

Symbols:: : solution, : sphero-conal coordinate, $\beta$ : sphero-conal coordinate, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Permalink:: http://dlmf.nist.gov/29.18.E4
Encodings:: TeX, pMML, png

where $u_{1}$ , $u_{2}$ , $u_{3}$ satisfy the differential equations

29.18.5 $\frac{d}{dr}\left(r^{2}\frac{du_{1}}{dr}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_% {1}=0,$

Symbols:: $\frac{df}{dx}$ : derivative of with respect to , $\nu$ : real parameter, $\omega$ : parameter, : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E5
Encodings:: TeX, pMML, png

29.18.6 $\frac{{d}^{2}u_{2}}{{d\beta}^{2}}+(h-\nu(\nu+1)k^{2}{\mathop{\mathrm{sn}\/}% \nolimits^{{2}}}\left(\beta,k\right))u_{2}=0,$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : real parameter, : real parameter, $\nu$ : real parameter, $\beta$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E6
Encodings:: TeX, pMML, png

29.18.7 $\frac{{d}^{2}u_{3}}{{d\gamma}^{2}}+(h-\nu(\nu+1)k^{2}{\mathop{\mathrm{sn}\/}% \nolimits^{{2}}}\left(\gamma,k\right))u_{3}=0,$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\frac{df}{dx}$ : derivative of with respect to , : real parameter, : real parameter, $\nu$ : real parameter, $\gamma$ : sphero-conal coordinate and $u_{j}$ : solutions
Referenced by:: §29.18(i)
Permalink:: http://dlmf.nist.gov/29.18.E7
Encodings:: TeX, pMML, png

with separation constants and $\nu$ . (29.18.5) is the differential equation of spherical Bessel functions (§10.47(i)), and (29.18.6), (29.18.7) agree with the Lamé equation (29.2.1).

§29.18(ii) Ellipsoidal Coordinates

Notes:: See Erdélyi et al. (1955, §15.1.3).
Keywords:: coordinate systems, ellipsoidal coordinates, wave equation
Referenced by:: §23.21(iii)
Permalink:: http://dlmf.nist.gov/29.18.ii

The wave equation (29.18.1), when transformed to ellipsoidal coordinates $\alpha,\beta,\gamma$ :

29.18.8

$x=k\mathop{\mathrm{sn}\/}\nolimits\left(\alpha,k\right)\mathop{\mathrm{sn}\/}% \nolimits\left(\beta,k\right)\mathop{\mathrm{sn}\/}\nolimits\left(\gamma,k% \right),$

$y=-\frac{k}{k^{{\prime}}}\mathop{\mathrm{cn}\/}\nolimits\left(\alpha,k\right)% \mathop{\mathrm{cn}\/}\nolimits\left(\beta,k\right)\mathop{\mathrm{cn}\/}% \nolimits\left(\gamma,k\right),$

$z=\frac{i}{kk^{{\prime}}}\mathop{\mathrm{dn}\/}\nolimits\left(\alpha,k\right)% \mathop{\mathrm{dn}\/}\nolimits\left(\beta,k\right)\mathop{\mathrm{dn}\/}% \nolimits\left(\gamma,k\right),$

with

29.18.9

$\alpha=\!\mathop{K\/}\nolimits\!+i\!\mathop{{K^{{\prime}}}\/}\nolimits\!-% \alpha^{{\prime}},$ $0\leq\alpha^{{\prime}}<\!\mathop{K\/}\nolimits\!$ ,

$\beta=\!\mathop{K\/}\nolimits\!+i\beta^{{\prime}},$ $0\leq\beta^{{\prime}}\leq 2\!\mathop{{K^{{\prime}}}\/}\nolimits\!,0\leq\gamma% \leq 4\!\mathop{K\/}\nolimits\!$ ,

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : real parameter, $\alpha$ : ellipsoidal coordinate, $\beta$ : ellipsoidal coordinate and $\gamma$ : ellipsoidal coordinate
Permalink:: http://dlmf.nist.gov/29.18.E9
Encodings:: TeX, TeX, pMML, pMML, png, png

admits solutions

29.18.10 $u(\alpha,\beta,\gamma)=u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma),$

Symbols:: $\alpha$ : ellipsoidal coordinate, $\beta$ : ellipsoidal coordinate and $\gamma$ : ellipsoidal coordinate
Permalink:: http://dlmf.nist.gov/29.18.E10
Encodings:: TeX, pMML, png

where $u_{1}$ , $u_{2}$ , $u_{3}$ each satisfy the Lamé wave equation (29.11.1).

§29.18(iii) Spherical and Ellipsoidal Harmonics

Notes:: See Arscott (1964b, §9.8.1), Erdélyi et al. (1955, §15.7), and Hobson (1931, Chapter XI).
Keywords:: Lamé polynomials, ellipsoidal harmonics, spherical harmonics
Permalink:: http://dlmf.nist.gov/29.18.iii

See Erdélyi et al. (1955, §15.7).

§29.18(iv) Other Applications

Keywords:: Lamé functions, conformal mapping
Permalink:: http://dlmf.nist.gov/29.18.iv

Triebel (1965) gives applications of Lamé functions to the theory of conformal mappings. Patera and Winternitz (1973) finds bases for the rotation group.

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 29.17 Other Solutions 29.19 Physical Applications

§29.18 Mathematical Applications

Contents

§29.18(i) Sphero-Conal Coordinates

§29.18(ii) Ellipsoidal Coordinates

§29.18(iii) Spherical and Ellipsoidal Harmonics

§29.18(iv) Other Applications