# §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 $\displaystyle x$ $\displaystyle=\frac{c\sinh\eta\cos\phi}{\cosh\eta-\cos\theta},$ $\displaystyle y$ $\displaystyle=\frac{c\sinh\eta\sin\phi}{\cosh\eta-\cos\theta},$ $\displaystyle z$ $\displaystyle=\frac{c\sin\theta}{\cosh\eta-\cos\theta},$

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

## §14.19(ii) Hypergeometric Representations

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

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

## §14.19(iii) Integral Representations

With $\xi>0$,

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

## §14.19(iv) Sums

With $\xi>0$,

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

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

With $\xi>0$,

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