Spherical harmonics are a useful tool in physics, particularly in classic electrostatics and electrodynamics. Given an integer $l$, the spherical harmonic $Y_{l,m}$, where $-l\leq m\leq l$, solves the PDE $$ \nabla^2Y_{l,m} + \frac{l(l+1)}{r^2}Y_{l,m} = 0 $$ Given $l$, there are $2l+1$ solutions to the above equation. These solutions are linearly independent (as the $Y_{l,m}$ form a complete set of orthonormal functions), but the degree of the equation is only 2, so there suppose to be only 2 linearly independent solutions that span the solution space. How is this not a contradiction?
2026-03-26 11:07:08.1774523228
Expressing spherical harmonics as a combination of other spherical harmonics
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