Dirac Delta Function

CORE DEFINITION

In science and mathematics, the Dirac delta function, or simply the delta function (also translated as Delta function), is a generalized function or distribution defined on the real line. It is zero everywhere except at zero, and its integral over the entire domain equals 1. The delta function can sometimes be viewed as an infinitely high and infinitely thin spike at the origin with total area 1, representing the density of an idealized point mass or point charge in physics. From a purely mathematical point of view, the Dirac delta function is not a function in the strict sense, because any function defined on the extended real line that is zero everywhere except at one point must have a total integral of zero. The delta function only has substantive meaning when it appears inside an integral. According to this, the delta function can generally be used like an ordinary function. The Dirac delta function is named after the physicist Paul Dirac, and the rules it formally obeys are part of operational calculus, a standard tool in physics and engineering. Operational calculus methods, including the delta function, were questioned by mathematicians in the early 20th century, and it was not until the 1950s that Laurent Schwartz developed a satisfactory rigorous theory.