# §14.19 Toroidal (or Ring) Functions

## §14.19(i) Introduction

When , , , and 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 , which are related to Cartesian coordinates by

where the constant is a scaling factor. Most required properties of toroidal functions come directly from the results for and . In particular, for and see §14.5(v).

## §14.19(ii) Hypergeometric Representations

With as in §14.3 and ,

14.19.2.

With ,