### The Maxima on-line user's manual

Algebra 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>.

`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) ```

### Related Examples

##### laplace-sin

laplace (H(t-3) * sin...

Calculate

##### laplace-plot2d-unit_step

f(t):= -2*(t-1.5)*uni...

plot2d(f(t),[t,0,5],[...

laplace(f(t),t,s);

Calculate

? laplace;

Calculate

##### laplace

laplace (1/(t*t),t,s);

Calculate

##### laplace

laplace (1/(s*(s^2+2*...

Calculate

##### laplace

inv_laplace(1/s(s^2+9...

Calculate

##### laplace-plot2d-unit_step

f(t):= t^2*unit_step(...

plot2d(f(t),[t,0,5],[...

laplace(f(t),t,s);

Calculate

##### laplace-sinh

laplace(t^(3/2)+sinh(...

Calculate

##### laplace

t^2;

laplace (%, t, s);

Calculate

##### laplace

inv_laplace(2/(s^2+s)...

Calculate