29 Lamé Functions Applications 29.17 Other Solutions 29.19 Physical Applications

§29.18 Mathematical Applications

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Referenced by:: §1.5(ii)
Permalink:: http://dlmf.nist.gov/29.18
See also:: Annotations for Ch.29

§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

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Keywords:: Lamé functions, applications, coordinate systems, separation constants, sphero-conal, sphero-conal coordinates, wave equation
Notes:: See Erdélyi et al. (1955, §15.1.2) and Jansen (1977).
Permalink:: http://dlmf.nist.gov/29.18.i
See also:: Annotations for §29.18 and Ch.29

The wave equation

29.18.1		$\nabla^{2}u+\omega^{2}u=0,$
ⓘ Symbols: $u$ : solution and $\omega$ : parameter Referenced by: §29.18(ii) Permalink: http://dlmf.nist.gov/29.18.E1 Encodings: TeX, pMML, png See also: Annotations for §29.18(i), §29.18 and Ch.29

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

29.18.2	$\displaystyle x$	$\displaystyle=kr\operatorname{sn}\left(\beta,k\right)\operatorname{sn}\left(% \gamma,k\right),$
	$\displaystyle y$	$\displaystyle=\mathrm{i}\frac{k}{k^{\prime}}r\operatorname{cn}\left(\beta,k% \right)\operatorname{cn}\left(\gamma,k\right),$
	$\displaystyle z$	$\displaystyle=\frac{1}{k^{\prime}}r\operatorname{dn}\left(\beta,k\right)% \operatorname{dn}\left(\gamma,k\right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $r$ : sphero-conal coordinate, $\beta$ : sphero-conal coordinate and $\gamma$ : sphero-conal coordinate Permalink: http://dlmf.nist.gov/29.18.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.18(i), §29.18 and Ch.29

with

29.18.3	$\displaystyle r$	$\displaystyle\geq 0,$
	$\displaystyle\beta$	$\displaystyle=K+\mathrm{i}\beta^{\prime},$
	$\displaystyle 0$	$\displaystyle\leq\beta^{\prime}\leq 2{K^{\prime}},$
	$\displaystyle 0$	$\displaystyle\leq\gamma\leq 4K,$
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : real parameter, $r$ : sphero-conal coordinate, $\beta$ : sphero-conal coordinate and $\gamma$ : sphero-conal coordinate Permalink: http://dlmf.nist.gov/29.18.E3 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §29.18(i), §29.18 and Ch.29

admits solutions

29.18.4		$u(r,\beta,\gamma)=u_{1}(r)u_{2}(\beta)u_{3}(\gamma),$
ⓘ Symbols: $u$ : solution, $r$ : 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 See also: Annotations for §29.18(i), §29.18 and Ch.29

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

29.18.5	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{\mathrm{d}u_{1}}{% \mathrm{d}r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1}$	$\displaystyle=0,$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $\nu$ : real parameter, $\omega$ : parameter, $r$ : sphero-conal coordinate and $u_{j}$ : solutions Referenced by: §29.18(i) Permalink: http://dlmf.nist.gov/29.18.E5 Encodings: TeX, pMML, png See also: Annotations for §29.18(i), §29.18 and Ch.29
29.18.6	$\displaystyle\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}}+(h-\nu(\nu+1)% k^{2}{\operatorname{sn}}^{2}\left(\beta,k\right))u_{2}$	$\displaystyle=0,$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : 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 See also: Annotations for §29.18(i), §29.18 and Ch.29
29.18.7	$\displaystyle\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1% )k^{2}{\operatorname{sn}}^{2}\left(\gamma,k\right))u_{3}$	$\displaystyle=0,$
ⓘ Symbols: $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $h$ : real parameter, $k$ : 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 See also: Annotations for §29.18(i), §29.18 and Ch.29

with separation constants $h$ 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

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Keywords:: coordinate systems, ellipsoidal, ellipsoidal coordinates, wave equation
Notes:: See Erdélyi et al. (1955, §15.1.3).
Referenced by:: §23.21(iii)
Permalink:: http://dlmf.nist.gov/29.18.ii
See also:: Annotations for §29.18 and Ch.29

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

29.18.8	$\displaystyle x$	$\displaystyle=k\operatorname{sn}\left(\alpha,k\right)\operatorname{sn}\left(% \beta,k\right)\operatorname{sn}\left(\gamma,k\right),$
	$\displaystyle y$	$\displaystyle=-\frac{k}{k^{\prime}}\operatorname{cn}\left(\alpha,k\right)% \operatorname{cn}\left(\beta,k\right)\operatorname{cn}\left(\gamma,k\right),$
	$\displaystyle z$	$\displaystyle=\frac{\mathrm{i}}{kk^{\prime}}\operatorname{dn}\left(\alpha,k% \right)\operatorname{dn}\left(\beta,k\right)\operatorname{dn}\left(\gamma,k% \right),$
ⓘ Symbols: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\mathrm{i}$ : imaginary unit, $x$ : real variable, $y$ : real variable, $z$ : complex variable, $k$ : real parameter, $\alpha$ : ellipsoidal coordinate, $\beta$ : ellipsoidal coordinate and $\gamma$ : ellipsoidal coordinate Permalink: http://dlmf.nist.gov/29.18.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §29.18(ii), §29.18 and Ch.29

with

29.18.9		$\displaystyle\alpha$	$\displaystyle=K+\mathrm{i}{K^{\prime}}-\alpha^{\prime},$
	$0\leq\alpha^{\prime}<K$ ,
		$\displaystyle\beta$	$\displaystyle=K+\mathrm{i}\beta^{\prime},$
	$0\leq\beta^{\prime}\leq 2{K^{\prime}},0\leq\gamma\leq 4K$ ,
ⓘ Symbols: ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathrm{i}$ : imaginary unit, $k$ : 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 See also: Annotations for §29.18(ii), §29.18 and Ch.29

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 See also: Annotations for §29.18(ii), §29.18 and Ch.29

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

§29.18(iii) Spherical and Ellipsoidal Harmonics

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Keywords:: Lamé polynomials, applications, ellipsoidal harmonics, spherical harmonics
Notes:: See Arscott (1964b, §9.8.1), Erdélyi et al. (1955, §15.7), and Hobson (1931, Chapter XI).
Permalink:: http://dlmf.nist.gov/29.18.iii
See also:: Annotations for §29.18 and Ch.29

See Erdélyi et al. (1955, §15.7).

§29.18(iv) Other Applications

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Keywords:: Lamé functions, applications, conformal mapping, rotation group
Permalink:: http://dlmf.nist.gov/29.18.iv
See also:: Annotations for §29.18 and Ch.29

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

29.17 Other Solutions 29.19 Physical Applications

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§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