Definition:Dirac Delta Function

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Definition

Definition 1

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:

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


Definition 2

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 < -\epsilon \\ \dfrac 1 {2 \epsilon } & : -\epsilon \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 Dirac delta function can be approximated as follows, where it is understood that the blue arrow represents a ray from $0$ up the $y$-axis:


Dirac-delta-function-limit.png


$2$ Dimensional Form

Let $\delta: \R \to \R$ denote the Dirac delta function.

The $2$-dimensional form of $\delta$ is defined and denoted:

$\forall x, y \in \R: \map { {}^2 \delta} {x, y} := \map \delta x \map \delta y$


Also denoted as

Let $c$ be a constant real number.

The notation $\map {\delta_c} t$ for the Dirac delta function is often used to denote:

$\map {\delta_c} t := \map \delta {t - c}$


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 known as

The Dirac delta function is less commonly rendered as Dirac's delta function.

It is also called the unit pulse function or unit impulse function.

Some sources refer to $\map \delta x$ just as the impulse function.

Some, acknowledging the fact that it is not actually a function as such, refer to it as the unit impulse.


Also defined as

The Dirac delta function is also defined by the following limits:

\(\text {(1)}: \quad\) \(\ds \map \delta x\) \(=\) \(\ds \dfrac 1 \pi \lim_{\epsilon \mathop \to 0} \dfrac \epsilon {x^2 + \epsilon^2}\)
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac 1 2 \lim_{\epsilon \mathop \to 0} \epsilon \size x^{\epsilon - 1}\)
\(\text {(3)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac 1 {\sqrt \pi} \lim_{\epsilon \mathop \to 0} \dfrac 1 {\sqrt {4 \epsilon} } e^{-x^2 / {4 \epsilon} }\)
\(\text {(4)}: \quad\) \(\ds \) \(=\) \(\ds \dfrac 1 {\pi x} \lim_{\epsilon \mathop \to 0} \map \sin {\dfrac x \epsilon}\)


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

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