Add patterns for the Laplace transform

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


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

The laplace function uses a set of patterns for computing Laplace transforms. You can extend the set by adding your own patterns. To add a new pattern to the pattern matcher, use laplace::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 laplace::addpattern can differ from the names that you use when calling laplace. 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 laplace::addpattern changes the expressions returned by future calls to laplace.


Example 1

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

laplace(foo(t), t, s)

Add a pattern for the Laplace transform of foo using laplace::addpattern:

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

Now laplace returns the Laplace transform of foo:

laplace(foo(t), t, s)

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

laplace(t^3 + exp(2*t)*foo(t), t, s)

Example 2

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

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

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

laplace(foo(t), t, s)

Example 3

Add this pattern for the Laplace transform of f:

laplace::addpattern(f(a*x)*g(a*x), x, y, y/(y^4 + 4*a^4)):
laplace(f(a*v)*g(a*v), v, w)

This pattern holds only when the first argument of f is the symbolic parameter a. If you use any other value of this parameter, laplace ignores the pattern:

laplace(f(A*v)*g(A*v), v, w)

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

laplace::addpattern(f(a*x)*g(a*x), x, y, y/(y^4 + 4*a^4), [a]):

Now laplace applies the specified pattern for an arbitrary value of a:

laplace(f(A*v)*g(A*v), v, w)

Example 4

Use assumptions when adding the following pattern for the Laplace transform:

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

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

laplace(FOO(x*t),t,s) assuming x >= 1

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

laplace(FOO(x*t), t, s)

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

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

If either the conditions are not satisfied or substituting the values into the result gives an error, laplace 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:

laplace(FOO(1/2*t), t, s)



Arithmetical expression in the variable t 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 s representing the pattern for the result of the transformation


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 t or s. 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