A Cauchy-Dirac delta function (Q360434)

In his famous book [The principles of quantum mechanics. 4th ed. London: Clarendon Press, Oxford University Press (1958; Zbl 0080.22005)], the physicist \textit{Paul Dirac} introduced a mathematical object that is nowadays called the Dirac delta function, and which satisfies the relations \(\int_{-\infty}^\infty\delta(x)\,{\mathrm d}x=1\), and \(\delta(x)=0\) for \(x\neq 0\). It is well known that the delta function is not a function in a strict mathematical sense, and that a rigorous definition was provided by the theory of distributions. Dirac was not the first person dealing with objects of a similar nature, and the present paper traces the history back to Augustin Cauchy. As the authors explain, his ideas involved infinitesimal quantities. For example, Cauchy arrived at the formula \[ \frac{1}{2}\int_{a-\varepsilon}^{a+\varepsilon}F(\mu)\frac{\alpha\,{\mathrm d}\mu}{\alpha^2+(\mu-a)^2}=\frac{\pi}{2}F(a), \] which holds for any function \(F\) whenever \(\varepsilon\) and \(\alpha\) are infinitely small numbers. Nowadays, the function \(\mu\mapsto \frac{1}{\pi}\frac{\alpha}{\alpha^2+(\mu-a)^2}\) is known as the probability density function of the Cauchy distribution. The authors also point the readers' attention to a different example due to Cauchy, where a delta function appears outside the integral sign. The rest of the paper discusses various topics related to infinitesimals and the process of their rigorization.