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 3D spherical case, which depend on
the two variables and .
© 19969 Eric W. Weisstein
19990525