Extended Siegert theorem

The extension of Siegert's theorem for the retarded electric multipole field is constructed. Analogously, a form of the electric current is developed which is manifestly conserved. Isolation of those components of the Fourier transform of the current which are constrained by current conservation is shown to be unique. In the new form of the electric multipole fields, the current enters only in the combination $\overrightarrow{\mu }\left(\overrightarrow{\mathrm{x}}\right)=\frac{1}{2}\overrightarrow{\mathrm{x}}\times \overrightarrow{\mathrm{J}}\left(\overrightarrow{\mathrm{x}}\right)$.