Re: Integrate piecewise with Assumptions

• To: mathgroup at smc.vnet.net
• Subject: [mg44153] Re: Integrate piecewise with Assumptions
• From: "Chia-Ming Yu" <yucalf at mail.educities.edu.tw>
• Date: Fri, 24 Oct 2003 04:24:30 -0400 (EDT)
• Organization: DCI HiNet
• References: <bn2iop\$o1e\$1@smc.vnet.net> <bn8der\$ndn\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Dear Sir Poujade,

You can call me Chia-Ming, I am a graduate student in Taiwan.
I believe that you have studied the machanism of integration for a period.

I believe they will help me a lot, and I will report the results of my test
to you as soon as possible.

Glad to discuss challenging questions with you!

Chia-Ming

news:bn8der\$ndn\$1 at smc.vnet.net ¤¤¼¶¼g...
> "Chia-Ming" <yucalf at mail.educities.edu.tw> wrote in message
news:<bn2iop\$o1e\$1 at smc.vnet.net>...
> > Excuse me,
> >
> > First, I define a piecewise function to discribe the shortest distance
> > on the edge of a circle.
> >
> > dis[x_, xi_] :=
> >   Which[1/2 >= x - xi >= 0, x - xi, x - xi > 1/2, 1 - (x - xi),
> >     0 > x - xi >= -1/2, xi - x, x - xi < -1/2, 1 - (xi - x)]
> >
> > And it does works that it can automatically judge to go in a clockwise
> > or a conter-clockwise direction.
> >
> > dis[2/3, 1/10]
> > 13/30
> >
> > Furthermore, dis[] can be applied in the Integrate[] command:
> >
> > Integrate[(a - 5 dist[1/6, x])^2, {x, 0, 1}]// Simplify
> > 25/12 - (5 a)/2 + a^2
> >
> > However, it can not work with the option Assumptions:
> >
> > Integrate[(a - 5 dist[y, x])^2, {x, 0, 1},
Assumptions->0<y<1/2]//Simplify
> [...]
>
> Dear Chia-Ming  (or should I write Dear Yu ?)
>
> You better use UnitStep functions to describe your function.
> to build piecewise constant functions,
> or piecewise linear functions, expressed
> in terms of UnitStep functions.
>
> In[1]:=(* 'pcf' is piecewise constant function constructor.
> Function is expressed in terms of UnitStep functions.
> call format : pcf[y0, listOfPoints, x]
> input :
>    y0 = leftmost value
>    listOfPoints = {{x1,y1},...,{xn,yn}} where  yi are right values
>    at points of discontiuity
>    x = variable used to generate output expression
> output :
>    piecewise constant function expressed in terms of UnitStep
> functions
>    using 'x' as variable
> *)
>
> pcf[y0_, listOfPoints_List, x_]:=
> Module[{lp = Length[listOfPoints], f, a, b, eq, vars},
>    f[u_] := Sum[a[i]UnitStep[u-listOfPoints[[i,1]]],{i,1,lp}]+b;
>    eq = Append[f[#[[1]]] == #[[2]]& /@ listOfPoints,f[-Infinity] ==
> y0];
>    vars = Append[Table[a[i],{i,1,lp}],b];
>    First[f[x] /. Solve[eq,vars]]
> ];
>
> (* example of a square bump *)
> pcf[0,{{0,1},{1,0}},x]
> Out[2]=-UnitStep[-1+x]+UnitStep[x]
>
> In[3]:=(* a stairs function : *)
> pcf[0,{{0,1},{1,2},{2,3},{3,4},{4,5},{5,0}},x]
> Out[3]=-5
UnitStep[-5+x]+UnitStep[-4+x]+UnitStep[-3+x]+UnitStep[-2+x]+UnitStep[-1+x]+
>   UnitStep[x]
>
> In[4]:=(* plf is a piecewise linear function constructor.
> (function may be discontinuous)
> call format : plf[leftSlope0_, listOfPoints_List, x_]
> input :
>    leftSlope0 : leftmost slope
>    listOfPoints format :
>    {{x1, lefty1, righty1, rightSlope1},...,
>     {xn, leftyn, rightyn, rightSlopen}}
>    x : variable
> output :
>    piecewise linear function expressed in terms of UnitStep functions
>    using 'x' as variable
> *)
>
> plf[leftSlope0_, listOfPoints_List, x_]:=
> Module[{deltaFunctions, pcfPoints, tempPlf, k, sol},
>    deltaFunctions = (#[[3]]-#[[2]])DiracDelta[x-#[[1]]]&/@listOfPoints;
>    pcfPoints = {#[[1]],#[[4]]}& /@ listOfPoints;
>    tempPlf =
Integrate[pcf[leftSlope0,pcfPoints,x]+Plus@@deltaFunctions,x]+k;
>    sol = Solve[(tempPlf/.x -> (listOfPoints[[1,1]]+
>
listOfPoints[[2,1]])/2)==(listOfPoints[[1,3]]+listOfPoints[[2,2]])/2,k];
>    tempPlf/.sol[[1]]
> ];
>
> (* example of a sawtooth function *)
> plf[0,{{0,0,0,1},{1,1,0,1},{2,1,0,1},{3,1,0,0}},x] //Simplify
>
> Out[5]=-(-2+x) UnitStep[-3+x]-UnitStep[-2+x]-UnitStep[-1+x]+x
> UnitStep[x]
>
> In[6]:=(* the bump function may be expressed two ways : *)
> plf[0,{{0,0,1,0},{1,1,0,0}},x] == pcf[0,{{0,1},{1,0}},x]//Simplify
> Out[6]=True
>