The core principle is that solutions to the Laplace equation in spherical coordinates can be constructed via separation of variables into radial functions and angular components defined by Legendre polynomials $P_n^m(\cos\theta)$ and complex exponentials, forming orthogonal Spherical Harmonics. This expansion relies on Sturm-Liouville theory principles where boundary conditions at singular points (the origin) enforce discrete eigenvalues for the angular momentum quantum number $n$ to ensure series convergence, while orthogonality relations allow decomposition of concentration or temperature fields into multipole moments corresponding to point sources ($l=0$), dipoles ($l=1$), and higher-order multipoles.
The core principle is that solutions to the Laplace equation in spherical coordinates can be constructed via separation of variables into radial functions and angular components defined by Legendre p…