Laplace's Equation--Toroidal Coordinates

In toroidal coordinates, Laplace's equation becomes

(1)

Attempt separation of variables by plugging in the trial solution

(2)

then divide the result by to obtain

(3)

The function then separates with

(4)

giving solution

(5)

Plugging back in and dividing by gives

(6)

The function then separates with

(7)

giving solution

(8)

For single-valued solutions that are -periodic functions of and , and are nonnegative integers. The modes and include constant factors. Plugging back in and multiplying by gives

(9)

which can also be written

(10)

(Arfken 1970, pp. 114-115). Laplace's equation is partially separable, although the Helmholtz differential equation is not.

Solutions to the differential equation for are known as toroidal functions.