# §29.18 Mathematical Applications

## §29.18(i) Sphero-Conal Coordinates

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 29

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

 29.18.2 $\displaystyle x$ $\displaystyle=kr\operatorname{sn}\left(\beta,k\right)\operatorname{sn}\left(% \gamma,k\right),$ $\displaystyle y$ $\displaystyle=\mathrm{i}\frac{k}{k^{\prime}}r\operatorname{cn}\left(\beta,k% \right)\operatorname{cn}\left(\gamma,k\right),$ $\displaystyle z$ $\displaystyle=\frac{1}{k^{\prime}}r\operatorname{dn}\left(\beta,k\right)% \operatorname{dn}\left(\gamma,k\right),$

with

 29.18.3 $\displaystyle r$ $\displaystyle\geq 0,$ $\displaystyle\beta$ $\displaystyle=\!K\!+\mathrm{i}\beta^{\prime},$ $\displaystyle 0$ $\displaystyle\leq\beta^{\prime}\leq 2\!{K^{\prime}}\!,$ $\displaystyle 0$ $\displaystyle\leq\gamma\leq 4\!K\!,$

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

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

 29.18.5 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}r}\left(r^{2}\frac{\mathrm{d}u_{1}}{% \mathrm{d}r}\right)+(\omega^{2}r^{2}-\nu(\nu+1))u_{1}$ $\displaystyle=0,$ 29.18.6 $\displaystyle\frac{{\mathrm{d}}^{2}u_{2}}{{\mathrm{d}\beta}^{2}}+(h-\nu(\nu+1)% k^{2}{\operatorname{sn}^{2}}\left(\beta,k\right))u_{2}$ $\displaystyle=0,$ 29.18.7 $\displaystyle\frac{{\mathrm{d}}^{2}u_{3}}{{\mathrm{d}\gamma}^{2}}+(h-\nu(\nu+1% )k^{2}{\operatorname{sn}^{2}}\left(\gamma,k\right))u_{3}$ $\displaystyle=0,$

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

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

 29.18.8 $\displaystyle x$ $\displaystyle=k\operatorname{sn}\left(\alpha,k\right)\operatorname{sn}\left(% \beta,k\right)\operatorname{sn}\left(\gamma,k\right),$ $\displaystyle y$ $\displaystyle=-\frac{k}{k^{\prime}}\operatorname{cn}\left(\alpha,k\right)% \operatorname{cn}\left(\beta,k\right)\operatorname{cn}\left(\gamma,k\right),$ $\displaystyle z$ $\displaystyle=\frac{\mathrm{i}}{kk^{\prime}}\operatorname{dn}\left(\alpha,k% \right)\operatorname{dn}\left(\beta,k\right)\operatorname{dn}\left(\gamma,k% \right),$

with

 29.18.9 $\displaystyle\alpha$ $\displaystyle=\!K\!+\mathrm{i}\!{K^{\prime}}\!-\alpha^{\prime},$ $0\leq\alpha^{\prime}<\!K\!$, $\displaystyle\beta$ $\displaystyle=\!K\!+\mathrm{i}\beta^{\prime},$ $0\leq\beta^{\prime}\leq 2\!{K^{\prime}}\!,0\leq\gamma\leq 4\!K\!$,

 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 29
where $u_{1}$, $u_{2}$, $u_{3}$ each satisfy the Lamé wave equation (29.11.1).