Complex source point solutions of the Helmholtz equation and the complex delta function: the missing link

Wavefields that are related to a source point with complex coordinates, which are not all real numbers, meanwhile have a long history. They have first been applied in electromagnetic theory, and subsequently also in acoustics. For example, such wavefields exhibit a directivity whose degree can be controlled by the imaginary part of the source point. The wavefields in question are obtained if in the fundamental solution for a real source point this point is replaced by its complex counterpart. With respect to the complex source point, sometimes it is mystically spoken about the associated complex delta function. In a rigorous way, however, only an equivalent source, which is defined on the real space, could be derived. In the present paper, we show that a wavefield, which is related to a complex source point and solves the Helmholtz equation, can be extended to an ultradistributional solution of the Helmholtz equation with the complex delta function as right-hand side. This result demonstrates that the connection between those wavefields and the complex delta function is not only a formal one. Our analysis is performed in one, two and three space dimensions.