Laplace Transform of Symbolic Expression
Compute the Laplace transform of
1/sqrt(x). By default, the transform is in terms of
syms x y f = 1/sqrt(x); laplace(f)
ans = pi^(1/2)/s^(1/2)
Specify Independent Variable and Transformation Variable
Compute the Laplace transform of
exp(-a*t). By default, the independent variable is
t, and the transformation variable is
syms a t f = exp(-a*t); laplace(f)
ans = 1/(a + s)
Specify the transformation variable as
y. If you
specify only one variable, that variable is the transformation variable. The
independent variable is still
ans = 1/(a + y)
Specify both the independent and transformation variables as
y in the second and third
ans = 1/(t + y)
Laplace Transforms of Dirac and Heaviside Functions
Compute the Laplace transforms of the Dirac and Heaviside functions.
syms t s syms a positive laplace(dirac(t-a),t,s)
ans = exp(-a*s)
ans = exp(-a*s)/s
Relation Between Laplace Transform of Function and Its Derivative
Show that the Laplace transform of the derivative of a function is expressed in terms of the Laplace transform of the function itself.
syms f(t) s Df = diff(f(t),t); laplace(Df,t,s)
ans = s*laplace(f(t), t, s) - f(0)
Laplace Transform of Array Inputs
Find the Laplace transform of the matrix
M. Specify the independent and transformation variables for
each matrix entry by using matrices of the same size. When the arguments are
laplace acts on them element-wise.
syms a b c d w x y z M = [exp(x) 1; sin(y) i*z]; vars = [w x; y z]; transVars = [a b; c d]; laplace(M,vars,transVars)
ans = [ exp(x)/a, 1/b] [ 1/(c^2 + 1), 1i/d^2]
laplace is called with both scalar and nonscalar
arguments, then it expands the scalars to match the nonscalars by using
scalar expansion. Nonscalar arguments must be the same size.
ans = [ x/a, 1/b^2] [ x/c, x/d]
Laplace Transform of Symbolic Function
Compute the Laplace transform of symbolic functions. When the first argument contains symbolic functions, then the second argument must be a scalar.
syms f1(x) f2(x) a b f1(x) = exp(x); f2(x) = x; laplace([f1 f2],x,[a b])
ans = [ 1/(a - 1), 1/b^2]
If Laplace Transform Cannot Be Found
laplace cannot transform the input
then it returns an unevaluated call.
syms f(t) s f(t) = 1/t; F = laplace(f,t,s)
F = laplace(1/t, t, s)
Return the original expression by using
ans = 1/t
f — Input
symbolic expression | symbolic function | symbolic vector | symbolic matrix
Input, specified as a symbolic expression, function, vector, or matrix.
var — Independent variable
t (default) | symbolic variable
Independent variable, specified as a symbolic variable. This variable is
often called the "time variable" or the "space variable." If you do not
specify the variable then, by default,
f does not contain
laplace uses the
symvar to determine the independent
transVar — Transformation variable
s (default) |
z | symbolic variable | symbolic expression | symbolic vector | symbolic matrix
Transformation variable, specified as a symbolic variable, expression,
vector, or matrix. This variable is often called the "complex frequency
variable." If you do not specify the variable then, by default,
s is the independent variable of
The Laplace transform F = F(s) of the expression f = f(t) with respect to the variable t at the point s is
If any argument is an array, then
laplaceacts element-wise on all elements of the array.
If the first argument contains a symbolic function, then the second argument must be a scalar.
To compute the inverse Laplace transform, use
The Laplace transform is defined as a unilateral or one-sided transform. This definition assumes that the signal f(t) is only defined for all real numbers t ≥ 0, or f(t) = 0 for t < 0. Therefore, for a generalized signal with f(t) ≠ 0 for t < 0, the Laplace transform of f(t) gives the same result as if f(t) is multiplied by a Heaviside step function.
For example, both of these code blocks:
syms t; laplace(sin(t))
syms t; laplace(sin(t)*heaviside(t))
1/(s^2 + 1).