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Add patterns for the inverse Laplace transform

MuPAD® notebooks are not recommended. Use MATLAB® live scripts instead.

MATLAB live scripts support most MuPAD functionality, though there are some differences. For more information, see Convert MuPAD Notebooks to MATLAB Live Scripts.


ilaplace::addpattern(pat, s, t, res, <vars, <conds>>)


ilaplace::addpattern(pat, s, t, res) teaches ilaplace to return ilaplace(pat, s, t) = res.

The ilaplace function uses a set of patterns for computing inverse Laplace transforms. You can extend the set by adding your own patterns. To add a new pattern to the pattern matcher, use ilaplace::addpattern. MuPAD® does not save custom patterns permanently. The new patterns are available in the current MuPAD session only.

Variable names that you use when calling ilaplace::addpattern can differ from the names that you use when calling ilaplace. See Example 2.

You can include a list of free parameters and a list of conditions on these parameters. These conditions and the result are protected from premature evaluation. This means that you can use not   iszero(a^2 - b) instead of hold( _not @ iszero )(a^2 - b).

The following conditions treat assumptions on identifiers differently:

  • a^2 - b <> 0 takes into account assumptions on identifiers.

  • not   iszero(a^2 - b) disregards assumptions on identifiers.

See Example 4.

Environment Interactions

Calling ilaplace::addpattern changes the expressions returned by future calls to ilaplace.


Example 1

Compute the inverse Laplace transform of the function bar. By default, MuPAD does not have a pattern for this function:

ilaplace(bar(s), s, t)

Add a pattern for the inverse Laplace transform of bar using ilaplace::addpattern:

ilaplace::addpattern(bar(s), s, t, foo(t)):

Now ilaplace returns the inverse Laplace transform of bar:

ilaplace(bar(s), s, t)

After you add a new transform pattern, MuPAD can use that pattern indirectly:

ilaplace(exp(-s)*bar(s), s, t)

Example 2

Define the inverse Laplace transform of bar(y) using the variables x and y as parameters:

ilaplace::addpattern(bar(y), y, x, foo(x)):

The ilaplace function recognizes the added pattern even if you use other variables as parameters:

ilaplace(bar(s), s, t)

Example 3

Add this pattern for the inverse Laplace transform of F:

ilaplace::addpattern(F(c, S)*G(c, S), S, T, T/(T^4 + 4*c^4)):
ilaplace(F(c, s)*G(c, s), s, t)

This pattern holds only when the first argument of F is the symbolic parameter c. If you use any other value of this parameter, ilaplace ignores the pattern:

ilaplace(F(A, s)*G(A, s), s, t)

To use the pattern for arbitrary values of the parameter, declare the parameter c as an additional pattern variable:

ilaplace::addpattern(F(c, S)*G(c, S), S, T, T/(T^4 + 4*c^4), [c]):

Now ilaplace applies the specified pattern for an arbitrary value of c:

ilaplace(F(C, s)*G(C, s), s, t)

You also can declare several parameters as pattern variables. For example, this pattern has two pattern variables, a and b:

ilaplace::addpattern(f(a*y + b), y, x, g(x/a - b), [a, b]):
ilaplace(f(2*s + B), s, t)

Example 4

Use assumptions when adding this pattern for the inverse Laplace transform:

ilaplace::addpattern(BAR(x*s), s, t, sin(1/(x - 1/2))*FOO(t),
                                       [x], [abs(x) < 1]):
ilaplace(BAR(x*s), s, t) assuming -1 < x < 1

If |x| ≥ 1, you cannot apply this pattern:

ilaplace(BAR(x*s), s, t) assuming x >= 1

If MuPAD cannot determine whether the conditions are satisfied, it returns a piecewise object:

ilaplace(BAR(x*s), s, t)

Note that the resulting expression defining the inverse Laplace transform of BAR(x*s) implicitly assumes that the value of x is not 1/2. A strict definition of the pattern is:

ilaplace::addpattern(BAR(x*t), s, t, sin(1/(x - 1/2))*FOO(t),
                     [x], [abs(x) < 1, x <> 1/2]):

If either the conditions are not satisfied or substituting the values into the result gives an error, ilaplace ignores the pattern. For this particular pattern, you can omit specifying the assumption x <> 1/2. If x = 1/2, MuPAD throws an internal “Division by zero.” error and ignores the pattern:

ilaplace(BAR(s/2), s, t)



Arithmetical expression in the variable s representing the pattern to match.


Identifier or indexed identifier used as a variable in the pattern


Identifier or indexed identifier used as a variable in the result


Arithmetical expression in the variable t representing a pattern for the result


List of identifiers or indexed identifiers used as “pattern variables” (placeholders in pat and res). You can use pattern variables as placeholders for almost arbitrary MuPAD expressions not containing s or t. You can restrict them by conditions given in the optional parameter conds.


List of conditions on the pattern variables

Return Values

Object of type DOM_NULL

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