# laplace

Laplace transform

## Description

example

laplace(f) returns the Laplace Transform of f. By default, the independent variable is t and the transformation variable is s.

example

laplace(f,transVar) uses the transformation variable transVar instead of s.

example

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

## Examples

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Compute the Laplace transform of 1/sqrt(x). By default, the transform is in terms of s.

syms x y
f = 1/sqrt(x);
laplace(f)
ans =
pi^(1/2)/s^(1/2)

Compute the Laplace transform of exp(-a*t). By default, the independent variable is t, and the transformation variable is s.

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 t.

laplace(f,y)
ans =
1/(a + y)

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

laplace(f,a,y)
ans =
1/(t + y)

Compute the Laplace transforms of 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

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)

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 nonscalars, 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]

If 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.

laplace(x,vars,transVars)
ans =
[ x/a, 1/b^2]
[ x/c,   x/d]

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 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 ilaplace.

ilaplace(F,s,t)
ans =
1/t

## Input Arguments

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

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, laplace uses t. If f does not contain t, then laplace uses the function symvar to determine the independent variable.

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, laplace uses s. If s is the independent variable of f, then laplace uses z.

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### Laplace Transform

The Laplace transform F = F(s) of the expression f = f(t) with respect to the variable t at the point s is

$F\left(s\right)=\underset{{0}^{–}}{\overset{\infty }{\int }}f\left(t\right)\text{\hspace{0.17em}}{e}^{-st}dt.$

## Tips

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

## Algorithms

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))

and

syms t;
laplace(sin(t)*heaviside(t))

return 1/(s^2 + 1).