In the context of working in spherical coordinates, we can generally find solutions to the Laplace equation by assuming they separate into radial and angular components (spherical harmonics). However there could also be solutions that obviously don’t separate (case in point the sum of two different separable solutions).
However, my question is: can ANY solution be represented as an infinite sum of these solutions? For example, in my physics textbook, it is assumed that if we assume azimuthal symmetry (m = 0) and we know what the potential is along the Z axis, then we can know the potential everywhere else in space. But this rests upon the fact that any solution in some volume D can always be represented by an infinite series of these separable solutions.
Is this true, and is this difficult to prove? Basically I’m asking because we always assume that this sum is the “general” solution, but how do we know that’s true? How do we know there’s not possibly other solutions that could be added?