This is a comment but I do not belong to the happy  few who have this privilege.
The exponential function (even with real arguments) is a  Schwartzian distribution.  The purpose of this comment is to document the fact that Schwartz showed that the Fourier transform can be extended to the space of ALL distributions (not just tempered ones as is in his seminal text) in 1952  (consult Wikipedia under  Paley-Wiener theorem, in particular,  Schwartzian Paley-Wiener theorem): His method establishes an isomorphism between the space of distributions on the line and a certain space of entire functions subject to suitable growth conditions.  (The precise details can be found in accessible form in Strichartz' book on Fourier transforms and distributions).  This allows a completely rigorous derivation of the above formula for the FT of such functions.  The easiest way to do this is to use the usual trick of first calculating the FT of the Dirac function (with complex singularity---there is no mystery about this---the Dirac "function" is a measure and so can be defined at any point even in a topological space) which follows immediately from the latter's filtering property.  One then uses the fact (which is valid also in this context) that the FT is its own inverse (modulo the usual games with constants).

I don't have the resources to check whether the Schwartz approach predates that of Gelfand and Silov  mentioned above.