A while ago I found a series of papers which do some wild magic with derivative operators:
- New Dirac Delta function based methods with applications to perturbative expansions in quantum field theory by Kempf/Jackson/Morales, 2014
- How to (Path-) Integrate by Differentiating also by Kempf/Jackson/Morales, 2015
- Integration by differentiation: new proofs, methods and examples by Jia/Tang/Kempf, 2016
The general theme is that evaluating functions on derivative operators , and applying this to delta functions , is occasionally useful and gives weird alternate characterizations of the Fourier transform and stuff like that.
The authors are physicists, unsurprisingly, and I’m sure there are a bunch of reasons why these results are either not that surprising or surprising-yet-not-useful, but I found them useful. However, when I revisited them trying to understand the ideas more closely, I found myself kinda overwhelmed and confused. So here’s a… totally different take, rederiving the main result by poking around.
tldr: the Fourier transform of is . Wait, what?