COMPLEX POINT SOURCE FOR THE 3D LAPLACE OPERATOR

The research about the so-called \emph{complex beams}, localized solutions of the Helmholtz wave equation, lead to the problem of finding the sources of such solutions, which may be formally expressed as a Dirac delta function of a complex argument. To investigate about the meaning of the Dirac delta distribution of complex argument, the Greenʹs function of the 3D Poisson problem with a point source localized at an imaginary position in free space is considered. The main physical features of the potential created by that source are described. The inverse problem consists in looking for the real source distribution which causes that potential. The sources appear on a disk in the real space. Their physical interpretation requires a regularization process based on including the border of the disk.