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General Solutions to the HHE

On ``spherical'' domains (the interior and exterior of a sphere, or in a spherical shell) the completely general solution to the HHE can therefore be written in stationary form as:

$\displaystyle \sum_L A_L J_L({\bf r}) + B_L N_L({\bf r})$ (13.38)

or (for scattering theory, mostly) in the outgoing wave form

$\displaystyle \sum_L C_L J_L({\bf r}) + S_L H^+_L({\bf r}).$ (13.39)

Inside a sphere, $ B_L$ and $ S_L$ must be zero. Outside a sphere, or in a spherical annulus, all the coefficients can be non-zero unlike the situation for the Laplace equation (why?).

[This should provoke deep thoughts about the fundamental significance of the Laplace equation. Are there any ``really'' stationary sources in the dynamical, covariant, universe? Do we expect to have a contribution to the zero frequency charge/current density distribution in any region of space? What would this correspond to?]

Robert G. Brown 2017-07-11