# §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

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

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

with

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

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

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

with

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

 29.18.10 $u(\alpha,\beta,\gamma)=u_{1}(\alpha)u_{2}(\beta)u_{3}(\gamma),$
where $u_{1}$, $u_{2}$, $u_{3}$ each satisfy the Lamé wave equation (29.11.1).