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The Maxima on-line user's manual

Algebra Calculator

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Laplace Calculator

Laplace

Function: laplace (<expr>, <t>, <s>) Attempts to compute the Laplace transform of <expr> with respect to the variable <t> and transform parameter <s>.

atvalue(y(x),x=0,4);
l:laplace(y(x),x,s);
eq:

laplace recognizes in <expr> the functions delta, exp, log, sin, cos, sinh, cosh, and erf, as well as derivative, integrate, sum, and ilt. If laplace fails to find a transform the function specint is called. specint can find the laplace transform for expressions with special functions like the bessel functions bessel_j, bessel_i, ... and can handle the unit_step function. See also specint.

If specint cannot find a solution too, a noun laplace is returned.

<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 ilt, 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
          (%i5) assume(a>0)$
          (%i6) laplace(gamma_incomplete(a,t),t,s),gamma_expand:true;
                                                        - a - 1
                                   gamma(a)   gamma(a) s
          (%o6)                    -------- - -----------------
                                      s            1     a
                                                  (- + 1)
                                                   s
          (%i7) factor(laplace(gamma_incomplete(1/2,t),t,s));
                                                        s + 1
                                sqrt(%pi) (sqrt(s) sqrt(-----) - 1)
                                                          s
          (%o7)                 -----------------------------------
                                          3/2      s + 1
                                         s    sqrt(-----)
                                                     s
          (%i8) assume(exp(%pi*s)>1)$
          (%i9) laplace(sum((-1)^n*unit_step(t-n*%pi)*sin(t),n,0,inf),t,s),simpsum;
                                   %i                         %i
                        ------------------------ - ------------------------
                                        - %pi s                    - %pi s
                        (s + %i) (1 - %e       )   (s - %i) (1 - %e       )
          (%o9)         ---------------------------------------------------
                                                 2
          (%i9) factor(%);
                                                %pi s
                                              %e
          (%o9)                   -------------------------------
                                                       %pi s
                                  (s - %i) (s + %i) (%e      - 1)

There are also some inexact matches for laplace. Try ?? laplace to see them.

(%o1)                                true
(%i2) 

Laplace Example

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