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)


 * Are you sure this is a formal definition? The mode of convergence is not specified, and I'm pretty sure it's not pointwise (which is suggested) but rather in the space of distributions. But this formal treatment requires considerable groundwork to get going.


 * If you're up for it: I have good memories to the book I was taught from, "Distributions: Theory and Applications" (https://epdf.pub/distributions-theory-and-applications.html), lectured by the late Duistermaat himself. &mdash; Lord_Farin (talk) 15:33, 8 June 2019 (EDT)