Distributions and Dirac delta functions

Abstract

The delta function was introduced by Paul Dirac as a generalization of the Kronecker delta to the continuous case. Initially it was defined heuristically as a “function” which equals zero everywhere except at zero, and equals infinity at zero, such that its integral equals to one. Clearly this object does not exist as a usual function, so for a long time it was just a convenient technical tool without any rigorous sense. Mathematical justification of formal manipulations with this function leads to extending the traditional concept of a function to so-called generalized functions of distributions, which play a fundamental role in modern Analysis. In this lecture I will try to give a brief introduction to the subject and consider some basic examples.