A Cauchy-Dirac Delta Function

Abstract

Abstract. The Dirac δ function has solid roots in 19th century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac's discovery by over a century, and illuminating the nature of Cauchy's infinitesimals and his infinitesimal definition of δ.

Approximate decoding for network coded inter-dependent data

“…The shape of decoding performance given ν Ã n (see Fig. 3 and Appendix A) is well captured by a cost model Cðν n ; ν Ã n Þ, referred to as the Cauchy-Dirac delta function [27], defined as…”

Rubik’s for Cryptographers

“…Rather, his approach to continuity was via what is known today as microcontinuity (see the subsection "Continuity"). Several recent articles, (Błaszczyk et al [14]; Borovik and Katz [16]; Bråting [20]; Katz and Katz [62], [64]; Katz and Tall [69]), have argued that a proto-Weierstrassian view of Cauchy is one-sided and obscures Cauchy's important contributions, including not only his infinitesimal definition of continuity but also such innovations as his infinitesimally defined ("Dirac") delta function, with applications in Fourier analysis and evaluation of singular integrals, and his study of orders of growth of infinitesimals that anticipated the work of Paul du Bois-Reymond, 8 Cantor's dubious claim that the infinitesimal leads to contradictions was endorsed by no less an authority than B. Russell; see footnote 15 in the subsection "Mathematical Rigor".…”

The classical theory of calculus of variations for generalized functions

“…This theory seems to be a good candidate, since it is an extension of classical distribution theory which allows to model nonlinear singular problems, while at the same time sharing many nonlinear properties with ordinary smooth functions, like the closure with respect to composition and several non trivial classical theorems of the calculus. One could describe generalized smooth functions as a methodological restoration of Cauchy-Dirac's original conception of generalized function, see [6,26,22]. In essence, the idea of Cauchy and Dirac (but also of Poisson, Kirchhoff, Helmholtz, Kelvin and Heaviside) was to view generalized functions as suitable types of smooth set-theoretical maps obtained from ordinary smooth maps depending on suitable infinitesimal or infinite parameters.…”

“…For example, the density of a Cauchy-Lorentz distribution with an infinitesimal scale parameter was used by Cauchy to obtain classical properties which nowadays are attributed to the Dirac delta, cf. [22].…”