# 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