This question is from [here](https://math.stackexchange.com/questions/2490828/exponential-derivative-of-delta-distribution). I'm asking it here as well to increase the number of people who see it and 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!



Exponential derivative of delta distribution?
Note that 
$$e^{a\partial/\partial x}f(x)=f(x+a)$$
is the translation operator, so your exponent of the delta function gives $2\pi \delta(x-in)$, which is indeed consistent with


$$\int\_{-\infty}^\infty e^{izy}dy=2\pi\delta(z)$$


for $z=x-in$.




---


All of this is purely formal, but there have been attempts to put it on a more secure ground by defining the delta function of a complex argument as an "ultradistribution". See for example [Theories of Generalised Functions](https://books.google.it/books?id=uOmiAgAAQBAJ&pg=121#v=onepage&q&f=false) (page 121) and [Distribution Theory and Transform Analysis](https://books.google.nl/books?hl=en&lr=&id=6wPLAgAAQBAJ&oi=fnd&pg=204) (page 204). *Caveat emptor.*




---


To avoid a possible confusion, there is also the definition of a delta function in the complex plane as $\delta^2(x+iy)=\delta(x)\delta(y)$, which is just the product of two delta functions of real argument. Here we are concerned with a single delta function $\delta(x+iy)$ of a complex argument. The two objects are contrasted in this [reference.](https://arxiv.org/abs/1605.04321)


