Re: convolution involving UnitStep

*To*: mathgroup at smc.vnet.net*Subject*: [mg126125] Re: convolution involving UnitStep*From*: Ulf-Dietrich Braumann <braumann at uni-leipzig.de>*Date*: Fri, 20 Apr 2012 07:42:32 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <jmogf6$46k$1@smc.vnet.net>*Reply-to*: braumann at uni-leipzig.de

Hi, while I was also ending up with a ConditionalExpression for t>1, then I tried to flip the functions under the integral due to assumend commutativity, but still got the same ConditionalExpression. For both cases, however, Mathematica 8 could plot in the interval t \[Element] [0,1]. Later I launched Mathematica 6, below you find the results: In[133]:= y[t_] := Integrate[h[t - s] g[s], {s, 0, t}]; y[t] // FullSimplify Out[134]= (-1 + Cos[1 - t]) UnitStep[-1 + t] - 2 (-1 + Cos[t]) UnitStep[t] In[135]:= y[t_] := Integrate[h[s] g[t - s], {s, 0, t}]; y[t] // FullSimplify Out[136]= \[Piecewise] { {1 + Cos[1 - t] - 2 Cos[t], t > 1}, {2 - 2 Cos[t], 0 < t <= 1} } No idea how to get Mathematica 8 to provide a piecewise functional solution for 0 < t <= 1 as well. Mathematica 7 btw. behaves also strange, even though the output is not restricted to t>1, the plots seem to be wrong. I also have tried to define g as g[t_]:= 2UnitBox[t-1/2]+UnitStep[t-1] but then the output of Integrate no longer can be simplified and given as \[Piecewise] defined function. Greetings - Ulf-Dietrich On Thu, 19 Apr 2012, J Davis wrote: > h[t_] = Sin[t]; > g[t_] = 2 UnitStep[t] - UnitStep[t - 1]; > y[t_] = Integrate[h[t-s]g[s],{s,0,t}] > > results in a conditional expression requiring t>1, but I want to > evaluate and plot t values from [0,1] as well as t>1. > > I tried HeavisideTheta as well as := in the definition of y to no > avail. Thanks for any help... > >