Distributions: What Exactly is the Dirac Delta “Function”?

The theory of distributions, also known as generalized functions, often remains something of an enigma to practicing physicists and engineers, even those working on heavily mathematical projects. This is especially true for the author, who despite the efforts of very capable instructors in graduate courses on distributions, Sobolev spaces, and the like, managed to get through without really internalizing the key points. The present article represents the fruits of my attempt, years later, to remedy the effects of my own laziness and stupidity. It is my…