Dirac delta function
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.
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)
dirac takes into account assumptions on
syms x real assumeAlso(x ~= 0) dirac(x)
ans = 0
For further computations, clear the assumptions on
recreating it using
Compute the Dirac delta function of
its first three derivatives.
Use a vector
n = [0,1,2,3] to specify the order of derivatives.
dirac function expands the scalar into a vector of the same
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)]
ans = [ Inf, -Inf, Inf, -Inf]
You can use
fplot to plot the Dirac delta
function over the default interval
[-5 5]. However,
x equal to
fplot does not plot the infinity.
Declare a symbolic variable
x and plot the symbolic
dirac(x) by using
syms x fplot(dirac(x))
To handle the infinity at
x equal to
numeric values instead of symbolic values. Set the
Inf value to
1 and plot the Dirac delta function by using
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, 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.
n— Order of derivative
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.
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
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
For complex values
x with nonzero imaginary parts,
dirac returns floating-point results for numeric
arguments that are not symbolic objects.
dirac acts element-wise on nonscalar inputs.
The input arguments
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.