Spherical delta functions and multipole expansions

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Abstract
The Cartesian–Taylor series for an analytic function in three dimensions is rewritten as a series of solid spherical harmonics. A discussion of the distribution theory definition of singular spherical harmonics is given, which leads to a definition of spherical delta functions. An expansion of source functions in spherical delta functions and their derivatives leads to multipole expansions for the fields which, in a distribution theory sense, are valid everywhere.