/sci/ - Science & Math

Pic related makes sense for [math]\rho[/math] of compact support, but it is well known that if you take the charge density formally as a dirac delta you also get the correct electric field. How the fuck is this justified rigorously? From what I have gathered, the dirac delta can at most be extended to a linear functional over [math]C^\infty[/math] put in here we have a smooth function with a singularity at x. If we take some regularization of the dirac delta, is the limit well defined?