Definition:Dirac Delta Function
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:
$2$ 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
- Definition:Kronecker Delta
- Definition:Heaviside Step Function
- Equivalence of Definitions of Dirac Delta Function
- Definition:Dirac Delta Distribution: A more rigorous definition
- Results about the Dirac delta function can be found here.
Source of Name
This entry was named for Paul Adrien Maurice Dirac.
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Impulse Functions. The Dirac Delta Function: $42$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Dirac delta function (delta function)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generalized function
- Weisstein, Eric W. "Delta Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeltaFunction.html