# [Maxima] More on enhanced laplace transforms

Sheldon Newhouse sen1 at math.msu.edu
Fri Mar 13 21:18:01 CDT 2009

```Sheldon Newhouse wrote:
> Hello,
>  I have developed a very primitive extension to the laplace transform
> routines to deal with the kinds of discontinuous functions often taught
> in basic ODE courses.
>
> The direct routine (i.e., taking the Laplace transform using some
> unit_step functions) is a simple wrapper around the 'specint' program in
> its use in Laplace transforms).
>
> The inverse routine is a pretty sloppy hack and only seems to work for
> linear combinations of polynomials and unit_step functions.
>
> Expressions involving  direct transforms including exponential and trig
> functions are just too complicated for the present version.  The
> transforms have to be massaged first to get rid of expressions like
> exp(A*s) in the denominators, etc.
>
> To save writing, the direct routine is called 'lap(f)' where f is a
> function of t, and
>  the inverse routine is ilap(F) where F is a function of s.
>
> Examples:
> (%i26) f: sum(unit_step(t-i)*(t+1)^i,i,1,3);
>
> (%o26) (t+1)*ustep(t-1)+(t+1)^2*ustep(t-2)+(t+1)^3*ustep(t-3)
> (%i27) lap(f);
>
> (%o27) %e^-(3*s)*((2*s^3+s^2)*%e^(2*s)+(9*s^3+6*s^2+2*s)*%e^s+64*s^3+48*s^2
>                                       +24*s+6)
>         /s^4
> (%i28) ilap(%);
>
> (%o28) (t-1)*ustep(t-1)+2*ustep(t-1)+(t-2)^2*ustep(t-2)+6*(t-2)*ustep(t-2)
>
> +9*ustep(t-2)+(t-3)^3*ustep(t-3)+12*(t-3)^2*ustep(t-3)
>                        +48*(t-3)*ustep(t-3)+64*ustep(t-3)
> (%i29) expand(%o28 -%o26);
>
> (%o29) 0
>
>
> I have put a file called 'My_laplace.mac"  on the web at
>   http://janus.math.msu.edu/sen/WWW/Maxima_Laplace
>
> in case anyone is interested.  (Note: this is new and probably has
> bugs.  All I can say is that I tested it with problems in some basic ODE
> books and it works on them).
>
> I would appreciate any comments which might improve the code and make,
> perhaps,  a useful research tool.
>
> -sen
>
>
>
>
>
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> .
>
>
Now, I put a new version on the web which also handles trig, exponential
functions and unit_steps.  It has some of my usual aliases, and is not
intended as a final version.

Suggestions, comments, bug reports, etc.  are welcome.

-sen
```