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SciMax Toolbox >> laplace

laplace

Maxima Function

Calling Sequence

laplace (expr, t, s)

Description

Attempts to compute the Laplace transform of expr with respect to the variable t and transform parameter s. If laplace cannot find a solution, a noun 'laplace is returned.

laplace recognizes in expr the functions delta, exp, log, sin, cos, sinh, cosh, and erf, as well as derivative, integrate, sum, and ilt. If some other functions are present, laplace may not be able to compute the transform.

expr may also be a linear, constant coefficient differential equation in which case atvalue of the dependent variable is used.

The required atvalue may be supplied either before or after the transform is computed. Since the initial conditions must be specified at zero, if one has boundary conditions imposed elsewhere he can impose these on the general solution and eliminate the constants by solving the general solution for them and substituting their values back.

laplace recognizes convolution integrals of the form integrate (f(x) * g(t - x), x, 0, t); other kinds of convolutions are not recognized.

Functional relations must be explicitly represented in expr; implicit relations, established by depends, are not recognized. That is, if f depends on x and y, f (x, y) must appear in expr.

See also , the inverse Laplace transform.

Examples:

(%i1) laplace (exp (2*t + a) * sin(t) * t, t, s);
                            a
                          %e  (2 s - 4)
(%o1)                    ---------------
                           2           2
                         (s  - 4 s + 5)
(%i2) laplace ('diff (f (x), x), x, s);
(%o2)             s laplace(f(x), x, s) - f(0)
(%i3) diff (diff (delta (t), t), t);
                          2
                         d
(%o3)                    --- (delta(t))
                           2
                         dt
(%i4) laplace (%, t, s);
                            !
               d            !         2
(%o4)        - -- (delta(t))!      + s  - delta(0) s
               dt           !
                            !t = 0
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