# Documentation

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# ilaplace

Inverse Laplace transform

## Syntax

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

## Description

ilaplace(F) returns the inverse Laplace transform of F using the default independent variable s for the default transformation variable t. If F does not contain s, ilaplace uses symvar.

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

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

## Input Arguments

 F Symbolic expression or function, vector or matrix of symbolic expressions or functions. var Symbolic variable representing the independent variable. This variable is often called the "complex frequency variable". Default: The variable s. If F does not contain s, then the default variable is determined by symvar. transVar Symbolic variable or expression representing the transformation variable. This variable is often called the "time variable". Default: The variable t. If t is the independent variable of F, then the default transformation variable is the variable x.

## Examples

Compute the inverse Laplace transform of this expression with respect to y at the transformation variable x:

syms x y
F = 1/y^2;
ilaplace(F, y, x)
ans =
x

Compute the inverse Laplace transform of this expression calling the ilaplace function with one argument. If you do not specify the independent variable, ilaplace uses the variable s.

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, ilaplace uses the variable t:

ilaplace(F)
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

If 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, ilaplace acts on them element-wise. If ilaplace is called with both scalar and nonscalar arguments, then ilaplace expands 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)]

collapse all

### Inverse Laplace Transform

The inverse Laplace transform is defined by a contour integral in the complex plane:

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

Here, c is a suitable complex number.

### Tips

• If F is a matrix, ilaplace acts element-wise on all components of the matrix.

• If transVar is a matrix, ilaplace acts element-wise on all components of the matrix.

• To compute the direct Laplace transform, use laplace.