§14.19 Toroidal (or Ring) Functions

When $\nu =n-\frac{1}{2}$, $n=0,1,2,\mathrm{\dots }$, $\mu \in ℝ$, and $x\in (1,\mathrm{\infty })$ 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 $(\eta ,\theta ,\varphi )$, which are related to Cartesian coordinates $(x,y,z)$ by

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

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

With $\xi >0$,

With $\xi >0$,

With $\xi >0$,