Legendre polynomials, spherical harmonics - from the ground up?

Question

I would like to explore these topics in depth, but I am at a loss as to where to start looking. I see a lot of people using them without even trying to understand them or rederive them. I'm not afraid of "complex mathematics", "advanced mathematics", "long and messy derivations" and what other excuse an author comes up to conceal his ignorance of the subject.

I thought that's why we're all in this, if it were simple, it wouldn't be interesting. So, I kindly ask of you fine gentlemen to point me to some nice literature that has perhaps allowed you a better insight into the topics of (associated) legendre polynomials, spherical harmonics (perhaps some correlations with the fourier series etc.) and some history to put things into context. Laplace, the gravitational potentials problem and other physical phenomena rigged through these tools.

Thank you!

Answer 1

Applied Analysis by the Hilbert Space Method (and the professor who wrote it) taught me everything I know about orthonormal basis sets of polynomials, including the Legendre, Hermite, Laguerre, associated Legendre, spherical harmonics, Bessels, Fourier series and transforms, as well as more general Hilbert space theory. I was fortunate enough to work from Prof. Holland's proofs (so I only had to pay copying costs), and he was a legendary teacher. I cannot think of a better book from which to learn this stuff.