## 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 3-D spherical case, which depend on the two variables and .