Symbolic expression, symbolic function, or vector or matrix of symbolic expressions or functions.
Symbolic variable representing the independent variable. This variable is often called the "time variable".
Default: The variable
Symbolic variable or expression representing the transformation variable. This variable is often called the "complex frequency variable".
Default: The variable
Compute the Laplace transform of this expression with respect
to the variable
x for the transformation variable
syms x y f = 1/sqrt(x); laplace(f, x, y)
ans = pi^(1/2)/y^(1/2)
Compute the Laplace transform of this expression calling the
with one argument. If you do not specify the independent variable,
syms a t y f = exp(-a*t); laplace(f, y)
ans = 1/(a + y)
If you also do not specify the transformation variable,
ans = 1/(a + s)
Compute the following Laplace transforms that involve the Dirac and Heaviside functions:
syms t s laplace(dirac(t - 3), t, s)
ans = exp(-3*s)
laplace(heaviside(t - pi), t, s)
ans = exp(-pi*s)/s
laplace cannot find an explicit representation
of the transform, it returns an unevaluated call:
syms f(t) s F = laplace(f, t, s)
F = laplace(f(t), t, s)
ilaplace returns the original expression:
ilaplace(F, s, t)
ans = f(t)
The Laplace transform of a function is related to the Laplace transform of its derivative:
syms f(t) s laplace(diff(f(t), t), t, s)
ans = s*laplace(f(t), t, s) - f(0)
Find the Laplace transform of this matrix. Use matrices of the same size to specify the independent variables and transformation variables.
syms a b c d w x y z laplace([exp(x), 1; sin(y), i*z],[w, x; y, z],[a, b; c, d])
ans = [ exp(x)/a, 1/b] [ 1/(c^2 + 1), 1i/d^2]
When the input arguments are nonscalars,
on them element-wise. If
laplace is called with
both scalar and nonscalar arguments, then
the scalar arguments into arrays of the same size as the nonscalar
arguments with all elements of the array equal to the scalar.
syms w x y z a b c d laplace(x,[x, w; y, z],[a, b; c, d])
ans = [ 1/a^2, x/b] [ x/c, x/d]
Note that nonscalar input arguments must have the same size.
When the first argument is a symbolic function, 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]
The Laplace transform is defined as follows: