Helmholtz Differential Equation--Spherical Surface

On the surface of a sphere, attempt separation of variables in spherical coordinates by writing

then the Helmholtz differential equation becomes

Dividing both sides by ,

which can now be separated by writing

The solution to this equation must be periodic, so must be an integer. The solution may then be defined either as a complex function

for , ..., , or as a sum of real sine and cosine functions

for , ..., . Plugging (4) into (3) gives

which is the Legendre differential equation for with

giving

Solutions are therefore Legendre polynomials with a complex index. The general complex solution is then

and the general real solution is

Note that these solutions depend on only a single variable . However, on the surface of a sphere, it is usual to express solutions in terms of the spherical harmonics derived for the three-dimensional spherical case, which depend on the two variables and .