An apparent paradox concerning the field of an ideal dipole

The electric or magnetic field of an ideal dipole is known to have a Dirac delta function at the origin. The usual textbook derivation of this delta function is rather ad hoc and cannot be used to calculate the delta-function structure for higher multipole moments. Moreover, a naive application of Gauss's law to the ideal dipole field appears to give an incorrect expression for the dipole's effective charge density. We derive a general result for the delta-function structure at the origin of an arbitrary ideal multipole field without using any advanced techniques from distribution theory. We find that the divergence of a singular vector field can contain a $\textit{derivative}$ of a Dirac delta function even if the field itself does not contain a delta function. We also argue that a physical interpretation of the delta function in the dipole field previously given in the literature is incorrect.