Thus the solutions are just the Legendre functions with .

A rational Legendre function of degree n is defined as:

These are the Legendre functions of the second kind, denoted by .

The first few associated Legendre functions, including those for negative values of m, are:

They are called the Legendre functions when defined in this more general way.

In general, of course, Legendre functions of non-integer order may also be included.

The Legendre functions of integer order are well known, but it may still be appropriate to note just the first few.

It may be noted that the Legendre functions of the second kind are all singular when.

All of the solutions considered so far have involved only Legendre functions of even order.

These Legendre functions are often referred to as toroidal harmonics.