Editing Dirac delta function (section)

===Infinitesimal delta functions===
[[Cauchy]] used an infinitesimal {{mvar|α}} to write down a unit impulse, infinitely tall and narrow Dirac-type delta function {{mvar|δ<sub>α</sub>}} satisfying <math display="inline">\int F(x)\delta_\alpha(x) \,dx = F(0)</math> in a number of articles in 1827.{{sfn|Laugwitz|1989}} Cauchy defined an infinitesimal in ''[[Cours d'Analyse]]'' (1827) in terms of a sequence tending to zero.  Namely, such a null sequence becomes an infinitesimal in Cauchy's and [[Lazare Carnot]]'s terminology.

[[Non-standard analysis]] allows one to rigorously treat infinitesimals. The article by {{harvtxt|Yamashita|2007}} contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the [[hyperreal number|hyperreals]].  Here the Dirac delta can be given by an actual function, having the property that for every real function {{mvar|F}} one has <math display="inline">\int F(x)\delta_\alpha(x) \, dx = F(0)</math> as anticipated by Fourier and Cauchy.