Inverse Laplace transform
Symbolic expression or function, vector or matrix of symbolic expressions or functions.
Symbolic variable representing the independent variable. This variable is often called the "complex frequency variable".
Default: The variable
Symbolic variable or expression representing the transformation variable. This variable is often called the "time variable".
Default: The variable
Compute the inverse Laplace transform of this expression with
y at the transformation variable
syms x y F = 1/y^2; ilaplace(F, y, x)
ans = x
Compute the inverse Laplace transform of this expression calling
ilaplace function with one argument. If you
do not specify the independent variable,
syms a s x F = 1/(s - a)^2; ilaplace(F, x)
ans = x*exp(a*x)
If you also do not specify the transformation variable,
ans = t*exp(a*t)
Compute the following inverse Laplace transforms that involve the Dirac and Heaviside functions:
syms s t ilaplace(1, s, t)
ans = dirac(t)
ilaplace(exp(-2*s)/(s^2 + 1) + s/(s^3 + 1), s, t)
ans = heaviside(t - 2)*sin(t - 2) - exp(-t)/3 +... (exp(t/2)*(cos((3^(1/2)*t)/2) + 3^(1/2)*sin((3^(1/2)*t)/2)))/3
ilaplace cannot find an explicit representation
of the transform, it returns an unevaluated call:
syms F(s) t f = ilaplace(F, s, t)
f = ilaplace(F(s), s, t)
laplace returns the original expression:
laplace(f, t, s)
ans = F(s)
Find the inverse 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 ilaplace([exp(x), 1; sin(y), i*z],[w, x; y, z],[a, b; c, d])
ans = [ exp(x)*dirac(a), dirac(b)] [ ilaplace(sin(y), y, c), dirac(1, d)*1i]
When the input arguments are nonscalars,
on them element-wise. If
ilaplace 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 ilaplace(x,[x, w; y, z],[a, b; c, d])
ans = [ dirac(1, a), x*dirac(b)] [ x*dirac(c), x*dirac(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; ilaplace([f1, f2],x,[a, b])
ans = [ ilaplace(exp(x), x, a), dirac(1, b)]
The inverse Laplace transform is defined by a contour integral in the complex plane:
Here, c is a suitable complex number.