# §14.19 Toroidal (or Ring) Functions

## §14.19(i) Introduction

When $\nu=n-\frac{1}{2}$, $n=0,1,2,\dots$, $\mu\in\mathbb{R}$, and $x\in(1,\infty)$ solutions of (14.2.2) are known as toroidal or ring functions. This form of the differential equation arises when Laplace’s equation is transformed into toroidal coordinates $(\eta,\theta,\phi)$, which are related to Cartesian coordinates $(x,y,z)$ by

 14.19.1 $\displaystyle x$ $\displaystyle=\frac{c\mathop{\sinh\/}\nolimits\eta\mathop{\cos\/}\nolimits\phi% }{\mathop{\cosh\/}\nolimits\eta-\mathop{\cos\/}\nolimits\theta},$ $\displaystyle y$ $\displaystyle=\frac{c\mathop{\sinh\/}\nolimits\eta\mathop{\sin\/}\nolimits\phi% }{\mathop{\cosh\/}\nolimits\eta-\mathop{\cos\/}\nolimits\theta},$ $\displaystyle z$ $\displaystyle=\frac{c\mathop{\sin\/}\nolimits\theta}{\mathop{\cosh\/}\nolimits% \eta-\mathop{\cos\/}\nolimits\theta},$

where the constant $c$ is a scaling factor. Most required properties of toroidal functions come directly from the results for $\mathop{P^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$ and $\mathop{\boldsymbol{Q}^{\mu}_{\nu}\/}\nolimits\!\left(x\right)$. In particular, for $\mu=0$ and $\nu=\pm\frac{1}{2}$ see §14.5(v).

## §14.19(ii) Hypergeometric Representations

With $\mathop{\mathbf{F}\/}\nolimits$ as in §14.3 and $\xi>0$,

14.19.2 $\mathop{P^{\mu}_{\nu-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits% \xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\mu\right)}{\pi% ^{1/2}\left(1-e^{-2\xi}\right)^{\mu}e^{(\nu+(1/2))\xi}}\*\mathop{\mathbf{F}\/}% \nolimits\!\left(\tfrac{1}{2}-\mu,\tfrac{1}{2}+\nu-\mu;1-2\mu;1-e^{-2\xi}% \right),$
$\mu\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\ldots$.
 14.19.3 $\mathop{\boldsymbol{Q}^{\mu}_{\nu-\frac{1}{2}}\/}\nolimits\!\left(\mathop{% \cosh\/}\nolimits\xi\right)=\frac{\pi^{1/2}\left(1-e^{-2\xi}\right)^{\mu}}{e^{% (\nu+(1/2))\xi}}\*\mathop{\mathbf{F}\/}\nolimits\!\left(\mu+\tfrac{1}{2},\nu+% \mu+\tfrac{1}{2};\nu+1;e^{-2\xi}\right).$

## §14.19(iii) Integral Representations

With $\xi>0$,

 14.19.4 $\displaystyle\mathop{P^{m}_{n-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+m+\frac{1}{2}\right)(% \mathop{\sinh\/}\nolimits\xi)^{m}}{2^{m}\pi^{1/2}\mathop{\Gamma\/}\nolimits\!% \left(n-m+\frac{1}{2}\right)\mathop{\Gamma\/}\nolimits\!\left(m+\frac{1}{2}% \right)}\*\int_{0}^{\pi}\frac{(\mathop{\sin\/}\nolimits\phi)^{2m}}{(\mathop{% \cosh\/}\nolimits\xi+\mathop{\cos\/}\nolimits\phi\mathop{\sinh\/}\nolimits\xi)% ^{n+m+(1/2)}}\mathrm{d}\phi,$ 14.19.5 $\displaystyle\mathop{\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\/}\nolimits\!\left(% \mathop{\cosh\/}\nolimits\xi\right)$ $\displaystyle=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+\frac{1}{2}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(n+m+\tfrac{1}{2}\right)\mathop{\Gamma\/}% \nolimits\!\left(n-m+\frac{1}{2}\right)}\*\int_{0}^{\infty}\frac{\mathop{\cosh% \/}\nolimits\!\left(mt\right)}{(\mathop{\cosh\/}\nolimits\xi+\mathop{\cosh\/}% \nolimits t\mathop{\sinh\/}\nolimits\xi)^{n+(1/2)}}\mathrm{d}t,$ $m.

## §14.19(iv) Sums

With $\xi>0$,

 14.19.6 $\mathop{\boldsymbol{Q}^{\mu}_{-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/% }\nolimits\xi\right)+2\sum_{n=1}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!% \left(\mu+n+\tfrac{1}{2}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\mu+\tfrac{% 1}{2}\right)}\mathop{\boldsymbol{Q}^{\mu}_{n-\frac{1}{2}}\/}\nolimits\!\left(% \mathop{\cosh\/}\nolimits\xi\right)\mathop{\cos\/}\nolimits\!\left(n\phi\right% )=\dfrac{\left(\frac{1}{2}\pi\right)^{1/2}\left(\mathop{\sinh\/}\nolimits\xi% \right)^{\mu}}{\left(\mathop{\cosh\/}\nolimits\xi-\mathop{\cos\/}\nolimits\phi% \right)^{\mu+(1/2)}},$ $\Re{\mu}>-\tfrac{1}{2}$.

## §14.19(v) Whipple’s Formula for Toroidal Functions

With $\xi>0$,

 14.19.7 $\mathop{P^{m}_{n-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi% \right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(n+m+\tfrac{1}{2}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(n-m+\tfrac{1}{2}\right)}\*\left(\frac{2}{\pi% \mathop{\sinh\/}\nolimits\xi}\right)^{1/2}\mathop{\boldsymbol{Q}^{n}_{m-\frac{% 1}{2}}\/}\nolimits\!\left(\mathop{\coth\/}\nolimits\xi\right),$
 14.19.8 $\mathop{\boldsymbol{Q}^{m}_{n-\frac{1}{2}}\/}\nolimits\!\left(\mathop{\cosh\/}% \nolimits\xi\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(m-n+\tfrac{1}{2}% \right)}{\mathop{\Gamma\/}\nolimits\!\left(m+n+\tfrac{1}{2}\right)}\*\left(% \frac{\pi}{2\mathop{\sinh\/}\nolimits\xi}\right)^{1/2}\mathop{P^{n}_{m-\frac{1% }{2}}\/}\nolimits\!\left(\mathop{\coth\/}\nolimits\xi\right).$