Definition:Dirac Delta Function/Warning

Warning about Dirac Delta Function

Note that while the Dirac delta function $\map \delta x$ is usually so referred to as a function and treated as a function, it is generally considered not actually to be a function at all.

Thus it is commonplace to see the following definition or derivation for the Dirac delta function:

$\map \delta x := \begin {cases} \infty & : x = 0 \\ 0 & : x \ne 0 \end {cases}$

While this can be considered as acceptable in the context of certain branches of engineering or physics, its use is not endorsed on $\mathsf{Pr} \infty \mathsf{fWiki}$ because of its lack of rigor.

For example, it is essential not only that the value of $\map \delta 0$ is not finite, but also that it is rigorously defined exactly how "not finite" it is.

That cannot be done without recourse to a definition using limits of some form.

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Special Functions: $\text {VIII}$. The Unit Impulse function or Dirac delta function