Definition:Dirac Delta Function/Definition 1

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Definition

Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Consider the real function $F_\epsilon: \R \to \R$ defined as:

$\map {F_\epsilon} x := \begin{cases} 0 & : x < 0 \\ \dfrac 1 \epsilon & : 0 \le x \le \epsilon \\ 0 & : x > \epsilon \end{cases}$


The Dirac delta function is defined as:

$\map \delta x := \ds \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$


Graph of Dirac Delta Function

The graph of the construction of the Dirac delta function is illustrated below:


Dirac-delta-function-1.png


Warning

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.


Also see

  • Results about the Dirac delta function can be found here.


Source of Name

This entry was named for Paul Adrien Maurice Dirac.


Sources