Helmholtz Differential Equation--Spherical Surface

On the surface of a Sphere, attempt Separation of Variables in Spherical Coordinates by writing

$$F(\theta, \phi) = \Theta (\theta)\Phi(\phi),$$ (1)

then the Helmholtz Differential Equation becomes

$${1\over\sin^2\phi}{d^2\Theta\over d\theta^2}\Phi + {\cos\phi... ...hi}\Theta + {d^2\Phi\over d\phi^2}\Theta + k^2\Theta\Phi = 0.$$ (2)

Dividing both sides by $\Phi\Theta$,

$$\left({{\cos\phi\sin\phi\over\Phi} {d\Phi\over d\phi }+ {\si... ...ft({{1\over \Theta }{d^2\Theta\over d\theta^2}+k^2}\right)= 0,$$ (3)

which can now be separated by writing

$${d^2\Theta\over d\theta^2}{1\over\Theta}= -(k^2+m^2).$$ (4)

The solution to this equation must be periodic, so $m$ must be an Integer. The solution may then be defined either as a Complex function

$$\Theta (\theta) = A_me^{i\sqrt{k^2+m^2}\,\theta}+B_me^{-i\sqrt{k^2+m^2}\,\theta}$$ (5)

for $m=-\infty$, ..., $\infty$, or as a sum of Real sine and cosine functions

$$\Theta(\theta) = S_m\sin(\sqrt{k^2+m^2}\,\theta)+C_m\cos(\sqrt{k^2+m^2}\,\theta)$$ (6)

for $m = 0$, ..., $\infty$. Plugging (4) into (3) gives

$${\cos\phi\sin\phi\over\Phi}{d\Phi\over d\phi} + {\sin^2\phi\over\Phi}{d^2\Phi\over d\phi^2} + m^2 = 0$$ (7)

$$\Phi'' + {\cos\phi\over\sin\phi}\Phi' + {m^2\over\sin^2\phi}\Phi = 0,$$ (8)

which is the Legendre Differential Equation for $x = \cos \phi$ with

$$m^2\equiv l(l+1),$$ (9)

giving

$$l^2+l-m^2=0$$ (10)

$$l={\textstyle{1\over 2}}(-1\pm \sqrt{1+4m^2}\,).$$ (11)

Solutions are therefore Legendre Polynomials with a Complex index. The general Complex solution is then

$$F(\theta,\phi)=\sum_{m=-\infty}^\infty P_l(\cos \phi)(A_me^{im\theta}+B_me^{-im\theta}),$$ (12)

and the general Real solution is

$$F(\theta,\phi)=\sum_{m=0}^\infty P_l(\cos\phi)[S_m\sin(m\theta)+C_m\cos(m\theta)].$$ (13)

Note that these solutions depend on only a single variable $m$. 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 $l$ and $m$.