Definition talk:Dirac Delta Function

This definition strikes me as informal. Vretblad always uses "function" in scare quotes when describing $\delta$, preferring to call it the "Dirac pulse", or the "Dirac distribution", and that it is a "generalization of the classical notion of a function". This is the way it was taught to me in my classes as well. Perhaps merge this under Definition:Dirac Measure as an informal definition. --GFauxPas (talk) 18:40, 12 November 2016 (EST)


 * Agreed. When I studied this in engineering a long time ago, we learned the formal definition as the limit as $\epsilon \to 0$ of:
 * $\delta \left({x}\right) = \begin{cases}

0 & : x < 0 \\ 0 & : x > \epsilon \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \end{cases}$
 * There may be details at the points $0$ and $\epsilon$ (that is, whether it is $<$ or $\le$ at whatever place). I have not got round to posting that up because I have not yet had the patience to work my way through any of my books to get to that point, and I am reluctant to post stuff up from memory because it is frequently wrong or untrustworthy or incomplete or all three. --prime mover (talk) 19:19, 12 November 2016 (EST)


 * My professor defined it as:


 * $\delta_\epsilon(t) = \dfrac{1}{\sqrt{2\pi\epsilon}}e^{-\frac{x^2}{2\epsilon}}$
 * where you allow $\epsilon \to 0^+$ inside an integral when such an integral converges. The definitions probably coincide in the sense that you can concoct an integral with them, of some sort, satisfying $\displaystyle \int_{a - \epsilon}^{a+\epsilon} f(x) \delta(x-a) \, \mathrm dx = f(a)$ --GFauxPas (talk) 20:07, 12 November 2016 (EST)


 * As is usual nowadays, I believe I am going senile, I am confused as to what I am expected to contribute here. If you have a rigorous definition (or more than one such definition), then post it (or them) up in the usual style. It remains to provide an equivalence proof. --prime mover (talk) 02:44, 13 November 2016 (EST)


 * I wasn't implying you should do anything, I'll hopefully do most of the work, though my schoolwork comes first. There are several definitions that need to be established before I can use Vretblad's exposition. --GFauxPas (talk) 15:38, 13 November 2016 (EST)

The formal definition of this has been implemented according to the source works on my own shelf. --prime mover (talk) 16:29, 29 May 2019 (EDT)