- Cauchy Problem. Fundamental Solutions
The Dirac delta function δ(x) is the singular generalized function (distribution) acting by the rule ∫ ∞
−∞ ϕ(x)δ(x) dx = ϕ(0)
for an arbitrary function ϕ(x) continuous at the point x = 0. The Dirac delta function plays an important role in the theory of linear PDEs. The
rigorous definition of this function as the limit of delta sequences of regular distributions, as well as its physical interpretation, can be found in Chapter 21 (see also the references therein).