Documentation

laplace

Laplace transform

Syntax

laplace(f)
laplace(f,transVar)
laplace(f,var,transVar)

Description

laplace(f) returns the Laplace transform of f using the default independent variable t and the default transformation variable s. If f does not contain t, fourier uses symvar.

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

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

Input Arguments

f

Symbolic expression, symbolic function, or vector or matrix of symbolic expressions or functions.

var

Symbolic variable representing the independent variable. This variable is often called the "time variable".

Default: The variable t. If f does not contain t, then the default variable is determined by symvar.

transVar

Symbolic variable or expression representing the transformation variable. This variable is often called the "complex frequency variable".

Default: The variable s. If s is the independent variable of f, then the default transformation variable is the variable z.

Examples

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

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 laplace function with one argument. If you do not specify the independent variable, laplace uses the variable t.

syms a t y
f = exp(-a*t);
laplace(f, y)
ans =
1/(a + y)

If you also do not specify the transformation variable, laplace uses the variable s:

laplace(f)
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

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

More About

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Laplace Transform

The Laplace transform is defined as follows:

F(s)=0f(t)estdt.

Tips

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

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

  • To compute the inverse Laplace transform, use ilaplace.

Introduced before R2006a

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