# dirac

Dirac delta function

## Syntax

``d = dirac(x)``
``d = dirac(n,x)``

## Description

example

````d = dirac(x)` represents the Dirac delta function of `x`.```

example

````d = dirac(n,x)` represents the `n`th derivative of the Dirac delta function at `x`.```

## Examples

### Handle Expressions Involving Dirac and Heaviside Functions

Compute derivatives and integrals of expressions involving the Dirac delta and Heaviside functions.

Find the first and second derivatives of the Heaviside function. The result is the Dirac delta function and its first derivative.

```syms x diff(heaviside(x), x) diff(heaviside(x), x, x)```
```ans = dirac(x) ans = dirac(1, x)```

Find the indefinite integral of the Dirac delta function. The results returned by `int` do not include integration constants.

`int(dirac(x), x)`
```ans = sign(x)/2```

Find the integral of the sine function involving the Dirac delta function.

```syms a int(dirac(x - a)*sin(x), x, -Inf, Inf)```
```ans = sin(a)```

### Use Assumptions on Variables

`dirac` takes into account assumptions on variables.

```syms x real assumeAlso(x ~= 0) dirac(x)```
```ans = 0```

For further computations, clear the assumptions on `x` by recreating it using `syms`.

`syms x`

### Evaluate Dirac Delta Function for Symbolic Matrix

Compute the Dirac delta function of `x` and its first three derivatives.

Use a vector `n = [0,1,2,3]` to specify the order of derivatives. The `dirac` function expands the scalar into a vector of the same size as `n` and computes the result.

```syms x n = [0,1,2,3]; d = dirac(n,x)```
```d = [ dirac(x), dirac(1, x), dirac(2, x), dirac(3, x)]```

Substitute `x` with `0`.

`subs(d,x,0)`
```ans = [ Inf, -Inf, Inf, -Inf]```

### Plot Dirac Delta Function

You can use `fplot` to plot the Dirac delta function over the default interval `[-5 5]`. However, `dirac(x)` returns `Inf` at `x` equal to `0`, and `fplot` does not plot the infinity.

Declare a symbolic variable `x` and plot the symbolic expression `dirac(x)` by using `fplot`.

```syms x fplot(dirac(x))```

To handle the infinity at `x` equal to `0`, use numeric values instead of symbolic values. Set the `Inf` value to `1` and plot the Dirac delta function by using `stem`.

```x = -1:0.1:1; y = dirac(x); idx = y == Inf; % find Inf y(idx) = 1; % set Inf to finite value stem(x,y)```

## Input Arguments

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Input, specified as a number, symbolic number, variable, expression, or function, representing a real number. This input can also be a vector, matrix, or multidimensional array of numbers, symbolic numbers, variables, expressions, or functions.

Order of derivative, specified as a nonnegative number, or symbolic variable, expression, or function representing a nonnegative number. This input can also be a vector, matrix, or multidimensional array of nonnegative numbers, symbolic numbers, variables, expressions, or functions.

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### Dirac Delta Function

The Dirac delta function, δ(x), has the value 0 for all x ≠ 0, and ∞ for x = 0. The Dirac delta function satisfies the identity

`$\underset{-\infty }{\overset{\infty }{\int }}\delta \left(x\right)\text{\hspace{0.17em}}dx=1\text{\hspace{0.17em}}\text{.}$`

This is a heuristic definition of the Dirac delta function. A rigorous definition of the Dirac delta function requires the theory of distributions or measure theory.

For any smooth function f and a real number a, the Dirac delta function has the property

`$\underset{-\infty }{\overset{\infty }{\int }}\delta \left(x-a\right)\text{ }f\left(x\right)=f\left(a\right)\text{\hspace{0.17em}}\text{.}$`

## Tips

• For complex values `x` with nonzero imaginary parts, `dirac` returns `NaN`.

• `dirac` returns floating-point results for numeric arguments that are not symbolic objects.

• `dirac` acts element-wise on nonscalar inputs.

• The input arguments `x` and `n` must be vectors or matrices of the same size, or else one of them must be a scalar. If one input argument is a scalar and the other one is a vector or a matrix, then `dirac` expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

## Version History

Introduced before R2006a