Re: Piecewise and Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg75277] Re: Piecewise and Integral*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>*Date*: Mon, 23 Apr 2007 05:42:14 -0400 (EDT)*Organization*: The Open University, Milton Keynes, UK*References*: <f09g01$pe$1@smc.vnet.net> <f0ekjv$prh$1@smc.vnet.net>

dh wrote: > $Version: 5.1 for Microsoft Windows (October 25, 2004) > > Hi, > > I am getting 10 as expected. Are you fooling yourself? > > Daniel Hi Daniel, Interesting. The issue might be version dependent since I have gotten the same result as Zosi, that is 5 rather 10 when lambda is equal to T/2 (See Out[13]), with Mathematica 5.2. In[1]:= $Version Out[1]= 5.2 for Microsoft Windows (June 20, 2005) In[2]:= T = 2; Ï? = T/4. f[x_] := Piecewise[{{0, Inequality[-T/2, LessEqual, x, Less, -Ï?]}, {10, -Ï? <= x <= Ï?}, {0, Inequality[Ï?, Less, x, LessEqual, T/2]}}] f[x_] := f[x - T] /; x > T/2 f[x_] := f[x + T] /; x < -T/2 Plot[f[x], {x, -2*T, 2*T}]; Î» = 0 a0 = (2/T)*Integrate[f[x], {x, -T/2. + Î», T/2. + Î»}] Î» = 1/10 a0 = (2/T)*Integrate[f[x], {x, -T/2. + Î», T/2. + Î»}] Î» = T/2 a0 = (2/T)*Integrate[f[x], {x, -T/2. + Î», T/2. + Î»}] Out[3]= 0.5 Out[8]= 0 Out[9]= 10. Out[10]= 1 -- 10 Out[11]= 10. Out[12]= 1 Out[13]= 5. > > > > zosi wrote: > >> Hi, > > >> I have a problem that I considered (and still consider) trivial. > > >> Let us suppose I have defined a Gate > >> (Period T and width \[Tau]= T/4), through Piecewise > >> and extended it periodically (see from In[1] to In[6]). > >> The Plot seems indicate that the extension is correct. > >> Now let us calculate the coefficient a0 according to > >> the usual formula In[13], having assumed that \[Lambda]=0 (In[12]) > >> The result Out[13] = 10 is correct. > >> > >> If I consider another case, i.e., I change the value of \[Lambda]=1/10, > >> the result, 10, is again correct (as expected !). > >> > >> But, if I try \[Lambda]= T/2 the result is 5 (instead of 10). Why ? > >> My interpretation is that the range > >> \[Tau]< x \[LessEqual] T/2 is not evaluated. > >> Infact, when I remove the ":" in In[3], the function is not defined > >> in the last interval ( but correctly Plotted !). > > >> Any hint to obtain 10 when \[Lambda]= T/2 ? > >> Thanks for your help > > >> G. Zosi > >> Dipartimento Fisica Generale > >> Universita di Torino > >> Italy > > >> ----------------- begin ----------------------- > > >> In[1]:= Clear["Global`*"] ; Remove["Global`*"];$Line=0; > > >> In[1]:= T=2; > > >> In[2]:= \[Tau]= T/4. > > >> In[3]:= f[x_] := Piecewise[ {{0, -T/2 \[LessEqual] x <- \[Tau]}, > >> {10, -\[Tau] \[LessEqual] x > >> \[LessEqual]\[Tau]}, > >> {0, \[Tau]< x \[LessEqual] T/2}}] > >> > >> In[4]:= f[x_]:=f[x-T]/;x >T/2 > > >> In[5]:= f[x_]:=f[x+T]/;x<-T/2 > > >> In[6]:= Plot[f[x],{x,-2 T,2 T}]; > > >> In[12]:= \[Lambda]=0 > > >> In[13]:= a0 = (2/T)*Integrate[f[x],{x,-T/2. +\[Lambda] ,T/2.+\[Lambda]}] > > >> Out[13] = 10 > > >> Now try with \[Lambda] = 1/10 and \[Lambda] = T/2 > >> > >> --------------------- end -------------------- > > > >