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# ilaplace

Inverse Laplace transform

## Syntax

ilaplace(F)
ilaplace(F,transVar)
ilaplace(F,var,transVar)

## Description

example

ilaplace(F) returns the Inverse Laplace Transform of F. By default, the independent variable is s and the transformation variable is t. If F does not contain s, ilaplace uses the function symvar.

example

ilaplace(F,transVar) uses the transformation variable transVar instead of t.

example

ilaplace(F,var,transVar) uses the independent variable var and transformation variable transVar instead of s and t, respectively.

## Examples

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Compute the inverse Laplace transform of 1/s^2. By default, the inverse transform is in terms of t.

syms s
F = 1/s^2;
ilaplace(F)
ans =
t

Compute the inverse Laplace transform of 1/(s-a)^2. By default, the independent and transformation variables are s and t, respectively.

syms a s
F = 1/(s-a)^2;
ilaplace(F)
ans =
t*exp(a*t)

Specify the transformation variable as x. If you specify only one variable, that variable is the transformation variable. The independent variable is still s.

syms x
ilaplace(F,x)
ans =
x*exp(a*x)

Specify both the independent and transformation variables as a and x in the second and third arguments, respectively.

ilaplace(F,a,x)
ans =
x*exp(s*x)

Compute the following inverse Laplace transforms that involve the Dirac and Heaviside functions:

syms s t
ilaplace(1,s,t)
ans =
dirac(t)
F = exp(-2*s)/(s^2+1);
ilaplace(F,s,t)
ans =
heaviside(t - 2)*sin(t - 2)

Find the inverse 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 nonscalars, ilaplace 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];
ilaplace(M,vars,transVars)
ans =
[        exp(x)*dirac(a),      dirac(b)]
[ ilaplace(sin(y), y, c), dirac(1, d)*1i]

If ilaplace 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.

syms w x y z a b c d
ilaplace(x,vars,transVars)
ans =
[ x*dirac(a), dirac(1, b)]
[ x*dirac(c),  x*dirac(d)]

If ilaplace cannot compute the inverse transform, then it returns an unevaluated call to ilaplace.

syms F(s) t
F(s) = exp(s);
f = ilaplace(F,s,t)
f =
ilaplace(exp(s), s, t)

Return the original expression by using laplace.

laplace(f,t,s)
ans =
exp(s)

Compute the Inverse 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;
ilaplace([f1 f2],x,[a b])
ans =
[ ilaplace(exp(x), x, a), dirac(1, b)]

## Input Arguments

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Input, specified as a symbolic expression, function, vector, or matrix.

Independent 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 ilaplace uses s. If F does not contain s, then ilaplace uses the function symvar to determine the independent variable.

Transformation variable, specified as a symbolic variable, expression, vector, or matrix. It is often called the "time variable" or "space variable." By default, ilaplace uses t. If t is the independent variable of F, then ilaplace uses x.

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### Inverse Laplace Transform

The inverse Laplace transform f = f(t) of F = F(s) is:

$f\left(t\right)=\frac{1}{2\pi i}\underset{c-i\infty }{\overset{c+i\infty }{\int }}F\left(s\right){e}^{st}ds.$

Here, c is a suitable complex number.

## Tips

• If any argument is an array, then ilaplace acts 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 direct Laplace transform, use laplace.