General aperture The wave that emerges from a point source has amplitude at location r that is given by the solution of the frequency domain wave equation for a point source (The Helmholtz Equation ), : where is the 3-dimensional delta function. Source: Internet
The density function of this distribution can be written as : The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion ). Source: Internet
Inserting an integral over theta: :: The derivative-factor ensures that popping the delta function in removes the integral. Source: Internet
Generally, when the term "Dirac delta function" is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Source: Internet
If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral. Source: Internet
In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function against a sufficiently "good" test function is. Source: Internet