Re: [Maxima-discuss] Integrating piecewise defined functions

[Barton W. <wi...@un...>]

Hi,

I think your examples are simple enough that you could do the calculations by hand.  Likely, that approach would give you the simplest answers;  for example, since f vanishes outside [0,1] and is one inside, we have

  integrate(f(t)*g(x-t), t, -2, +2) = integrate(g(x-t), t, 0,1)

Now all you need is an antiderivative of g and you are done. And an antiderivative of g can be found by integrating each piece and choosing the constants of integration to make a continuous function.

 But maybe your aim is to integrate some more complicated piecewise defined functions.  Maxima has a share package that attempts to integrate expressions that can be expressed in terms of signum functions. For your examples:

(%i1) load(abs_integrate)$

(%i2) f(x) := (signum(x) - signum(x-1))/2$

(%i3) g(x) :=  x*f(x)$


(%i4) integrate(f(x),x,-2,4);
(%o4) 1

(%i5) integrate(f(t)*g(x-t), t, -2, 2);
(%o5) (x^2*signum(x)-3*signum(x-1)*x^2+2*signum(x-2)*x^2+4*signum(x-1)*x-4*signum(x-2)*x+signum(x-1)*(x-1)^2-signum(x-2)*(x-1)^2-
signum(x-1)+signum(x-2))/4

Let us know if you have questions.  But I have a turkey and a sweet potato casserole to bake this morning, so till later...

--Barton

________________________________
From: Jaime Villate via Maxima-discuss <max...@li...>
Sent: Thursday, November 28, 2024 01:51
To: Eduardo Ochs <edu...@gm...>; max...@li... <max...@li...>
Subject: Re: [Maxima-discuss] Integrating piecewise defined functions

Caution: Non-NU Email



On 27/11/24 21:56, Eduardo Ochs wrote:
Hi list,

what is the recommended way to integrate piecewise defined
functions? I tried this,

  f(x) := if     x<0 then 0
          elseif x<1 then 1
          else            0;
  g(x) := if     x<0 then 0
          elseif x<1 then x
          else            0;
  h(x) := integrate(f(t)*g(x-t), t, -2, +2);

and I couldn't find a way to make Maxima calculate this
integral...

Hello Eduardo,

If you mean the convolution of f and g:

   h(x) := integrate(f(x)*g(x-t), t, 0, x)

I recommend that you use Laplace transforms and expressions, rather than "functions":

f: hstep (x) - hstep (x-1);
g: (hstep (x) - hstep (x-1))*x;
F: laplace (f,x,s);
G: laplace (g,x,s);
H: F*G;
h: pwilt (H,s,x);
plot2d ([f,g,h], [x,-2,2], [legend,"f(x)","g(x)","h(x)"]);


Cumprimentos,

Jaime