Dirac delta function
dirac(x)
dirac(n,x)
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 this expression involving the Dirac delta function.
syms a int(dirac(x - a)*sin(x), x, -Inf, Inf)
ans = sin(a)
dirac
takes into account
assumptions on variables.
syms x real assumeAlso(x ~= 0) dirac(x)
ans = 0
For further computations, clear the assumptions.
syms x clear
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.
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]
To handle the infinity at 0, use numeric values instead of
symbolic values. Continue plotting all other symbolic inputs symbolically by
using fplot
.
Set the Inf
value to 1
and plot 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)

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.
At least one input argument must be a scalar or both
arguments must be vectors or matrices of the same size. 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.