Infinitesimal delta functions Cauchy used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δα satisfying in a number of articles in 1827. 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 contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the hyperreals. Here the Dirac delta can be given by an actual function, having the property that for every real function F one has as anticipated by Fourier and Cauchy. Dirac comb A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Sha distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense, which is a sequence of point masses at each of the integers. Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if is any Schwartz function, then the periodization of is given by the convolution In particular, is precisely the Poisson summation formula. More generally, this formula remains to be true if is a tempered distribution of rapid descent or, equivalently, if is a slowly growing, ordinary function within the space of tempered distributions. Sokhotski–Plemelj theorem The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution p.v. 1/x, the Cauchy principal value of the function 1/x, defined by Sokhotsky's formula states that Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f, Relationship to the Kronecker delta The Kronecker delta δij is the quantity defined by for all integers i, j. This function then satisfies the following analog of the sifting property: if is any doubly infinite sequence, then Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function. Applications Probability theory In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or