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Re: Integration of piecewise function

• To: mathgroup at smc.vnet.net
• Subject: [mg52485] Re: [mg52463] Integration of piecewise function
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Mon, 29 Nov 2004 01:22:32 -0500 (EST)
• Reply-to: hanlonr at cox.net
• Sender: owner-wri-mathgroup at wolfram.com

Define the function with UnitStep

Clear[X,w,c];

X[w_] := UnitStep[w]-2UnitStep[w-2]+UnitStep[w-4];

Plot[X[w],{w,-1,5}, PlotStyle->RGBColor[1, 0, 0]];

At the discontinuities, the limit is from the right

X /@ {0,2,4}

{1,-1,0}

Plot[Evaluate[Integrate[X[t], {t,-Infinity,w}]],{w,-1,5}, 
    PlotStyle->RGBColor[1, 0, 0]];

Integrate[X[w],{w,-Infinity,Infinity}]

0

More generally, for the integral from w1 to w2 with w1<=w2

int = Integrate[X[w],{w,-Infinity,w2}]-
    Integrate[X[w],{w,-Infinity,w1}]

(-(w1 - 4))*UnitStep[w1 - 4] + 2*(w1 - 2)*UnitStep[w1 - 2] - w1*UnitStep[w1] 
+ (w2 - 4)*UnitStep[w2 - 4] - 
  2*(w2 - 2)*UnitStep[w2 - 2] + w2*UnitStep[w2]

Note that the integral is evaluated as the difference of two integrals.  This 
avoids problems in evaluating the results at the discontinuities such as

Integrate[X[w],{w,w1,w2}]/. 
  {{w1->0, w2->8},{w1->2, w2->8},{w1->4, w2->8}}

{8,-14,4}

The actual values are

int /. {{w1->0, w2->8},{w1->2, w2->8},{w1->4, w2->8}}

{0,-2,0}

Cases for different ranges of w1 and w2 (w1 <= w2) can be evaluated using 
Simplify

c[w_] := (#[[1]]<=w<#[[2]])& /@ 
      Partition[{-Infinity,0,2,4,Infinity},2,1];

cond = And@@#&/@Simplify[
        Select[
          Flatten[
            Outer[List,c[w1],c[w2]],1],
          #[[1,1]]<=#[[2,1]]&], 
        Element[{w1,w2}, Reals]];

Simplify[Table[
      {Simplify[int,cond[[n]]],cond[[n]]},
      {n,Length[cond]}] //. 
    {s___,{0, c1_}, m___,{0,c2_},f___}:> 
      {s,{0,c1||c2},m,f}]

{{0, (w2 >= 4 && (w1 >= 4 || w1 < 0)) || (w1 < 0 && w2 < 0)}, 
  {w2, w1 < 0 && Inequality[0, LessEqual, w2, Less, 2]}, 
  {4 - w2, w1 < 0 && Inequality[2, LessEqual, w2, Less, 4]}, 
  {w2 - w1, Inequality[0, LessEqual, w1, Less, 2] && Inequality[0, LessEqual, 
w2, Less, 2]}, 
  {-w1 - w2 + 4, Inequality[0, LessEqual, w1, Less, 2] && Inequality[2, 
LessEqual, w2, Less, 4]}, 
  {-w1, Inequality[0, LessEqual, w1, Less, 2] && w2 >= 4}, 
  {w1 - w2, Inequality[2, LessEqual, w1, Less, 4] && Inequality[2, LessEqual, 
w2, Less, 4]}, 
  {w1 - 4, Inequality[2, LessEqual, w1, Less, 4] && w2 >= 4}}

Similar results are available much more directly in version 5.1 using 
PiecewiseExpand

Simplify[PiecewiseExpand[int], w1<=w2]

Piecewise[{{w1 - 4, Inequality[2, LessEqual, w1, Less, 4] && w2 >= 4}, 
   {-w1, Inequality[0, LessEqual, w1, Less, 2] && w2 >= 4}, 
   {4 - w2, w1 < 0 && Inequality[2, LessEqual, w2, Less, 4]}, 
   {-w1 - w2 + 4, Inequality[0, LessEqual, w1, Less, 2] && Inequality[2, 
LessEqual, w2, Less, 4]}, 
   {w1 - w2, Inequality[2, LessEqual, w1, Less, 4] && Inequality[2, LessEqual, 
w2, Less, 4]}, 
   {w2, w1 < 0 && Inequality[0, LessEqual, w2, Less, 2]}, 
   {w2 - w1, Inequality[0, LessEqual, w1, Less, 2] && Inequality[0, LessEqual, 
w2, Less, 2]}}]

However, note that one case is missing when using PiecewiseExpand.


Bob Hanlon

> 
> From: Riccardo <ricky at nospam.it>
To: mathgroup at smc.vnet.net
> Date: 2004/11/28 Sun AM 01:06:46 EST
> To: mathgroup at smc.vnet.net
> Subject: [mg52485] [mg52463] Integration of piecewise function
> 
> <inviati & spediti>
> 
> I just defined a function like this:
> 
> X[\[Omega]_] := 0 /; \[Omega] < 0;
> X[\[Omega]_] := 1 /; 0 < \[Omega] < 2;
> X[\[Omega]_] := -1 /; 2 < \[Omega] < 4;
> X[\[Omega]_] := 0 /; \[Omega] > 4;
> 
> Is it possible now to integrate it from -infinity to +infinity? I've
> already tried and it doesn't give me the result.
> Thanks.
> 
> Luca
> 
>