Helmholtz Differential Equation--Spherical Surface

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

(1)

then the Helmholtz differential equation becomes

(2)

Dividing both sides by ,

(3)

which can now be separated by writing

(4)

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

(5)

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

(6)

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

(7)

(8)

which is the Legendre differential equation for  with

(9)

giving

(10)

(11)

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

(12)

and the general real solution is

(13)

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 .