need help with integration

• To: mathgroup at smc.vnet.net
• Subject: [mg51997] need help with integration
• From: "symbio" <symbio at has.com>
• Date: Sat, 6 Nov 2004 02:08:53 -0500 (EST)
• Reply-to: "symbio" <symbio at sha.com>
• Sender: owner-wri-mathgroup at wolfram.com

Below I define two functions x[t] and h[t], then in eq1 I integrate
the integrand ( x[tau] h[t-tau] ) from -inf to +inf and get one set of
results and plots, then I integrate the same integrand as before but this
time in two steps, once from -inf to 0 and once from 0 to +inf, but the
results and plots from the integration performed in two steps are NOT the
same as the results and plots from integration in one step.  Is this a bug
and if so what's the work around?

Try this in the input cell

In[19]:=
x[t_] = (3Exp[-.5t]UnitStep[t]) + DiracDelta[t + 3]
h[t_] = UnitStep[t] - UnitStep[t - 2]
eq1 = Integrate[x[\[Tau]]h[t - \[Tau]], {\[Tau], -\[Infinity], \[Infinity]}]
Plot[eq1, {t, -10, 10}, PlotRange -> All]
eq2 = Integrate[x[\[Tau]]h[t - \[Tau]], {\[Tau], 0, \[Infinity]}]
Plot[eq2, {t, -10, 10}, PlotRange -> All]
eq3 = Integrate[x[\[Tau]]h[t - \[Tau]], {\[Tau], -\[Infinity], 0}]
Plot[eq3, {t, -10, 0}, PlotRange -> All]

You will get the following output expressions (I couldn't paste the plots
here, but if you run the above input cells you should get the plots too,
then it will be more obvious what the problem is)

Out[21]=
\!\(1\/2\ \((1 + \(6.`\ \@\((\(-2\) + t)\)\^2\)\/\(\(\(2.`\)\(\
\[InvisibleSpace]\)\) - 1.`\ t\) + \(6.`\ \@t\^2\)\/t + \@\((3 +
t)\)\^2\/\(3 \
+ t\) - \(1 + t + \@\((1 + t)\)\^2\)\/\(1 + t\))\)\)

Out[23]=
\!\(\((\(-6.`\) +
16.30969097075427`\ \[ExponentialE]\^\(\(-0.5`\)\ t\))\)\ UnitStep[\
\(-2\) + t] + \((\(\(6.`\)\(\[InvisibleSpace]\)\) -
6.`\ \[ExponentialE]\^\(\(-0.5`\)\ t\))\)\ UnitStep[t]\)

Out[25]=
\!\(1\/2\ \((\(-\(\(1 +
t\)\/\@\((1 + t)\)\^2\)\) + \(3 + t\)\/\@\((3 + t)\)\^2)\)\)

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