This question is essentially this one, but uses a different phrasing that hopefully might attract the attention of different people who might be able to help.
The question is, what is the result of the following integral for integer $n$ and real $x$?
$$\int_{-\infty}^\infty dy\, e^{ny}e^{iyx}$$
Does it diverge and give infinity, or is it actually just equal to the following?
$$=\sum_{m=0}^\infty\frac{n^m}{m!}\int_{-\infty}^\infty dy\, y^m e^{iyx}=2\pi\sum_{m=0}^\infty\frac{1}{m!}\left(-in\frac{\partial}{\partial x}\right)^m\delta(x)=2\pi \left(e^{-in\frac{\partial}{\partial x}}\delta(x)\right)$$
More importantly, is there some book or scientific article on distributions that discusses this? Thanks for any suggestion!