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Integration of UnitStep has bugs!? help!
*To*: mathgroup at smc.vnet.net
*Subject*: [mg51578] Integration of UnitStep has bugs!? help!
*From*: "news" <symbio at s.dn.com>
*Date*: Sat, 23 Oct 2004 00:22:21 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Can anyone explain why Integrate does not work propery with UnitStep
function? When you integrate from -inf to inf you get one answer, but when
you split it to two sections (from -inf to 0, and then from 0 to +inf) you
get a different answer. Which is correct? And why is Mathematica version 5
has not fixed this problem yet? See below.
In[1]:=
\!\(\(x[t_] = \((3 Exp[\(\(-1\)\/2\) t] UnitStep[t])\) +
DiracDelta[t + 3];\)\[IndentingNewLine]
\(h[t_] = UnitStep[t] - UnitStep[t - 2];\)\[IndentingNewLine]
\(eq1 =
Integrate[
x[a] h[t -
a], {a, \(-\[Infinity]\), \[Infinity]}];\)\[IndentingNewLine]
\(eq2 = Integrate[x[a] h[t - a], {a, 0, \[Infinity]}];\)\[IndentingNewLine]
\(eq3 =
Integrate[
x[a] h[t - a], {a, \(-\[Infinity]\), 0}];\)\[IndentingNewLine]
eq1\ // FullSimplify\[IndentingNewLine]
eq2 + eq3 // FullSimplify\)
Out[6]=
\!\(1\/2\ \((\(-\(\(6\ \((\(-2\) +
t)\)\)\/\@\((\(-2\) + t)\)\^2\)\) + \(6\ t\)\/\@t\^2 - \
\(1 + t\)\/\@\((1 + t)\)\^2 + \(3 + t\)\/\@\((3 + t)\)\^2)\)\)
Out[7]=
\!\(1\/2\ \((\(-\(\(1 +
t\)\/\@\((1 + t)\)\^2\)\) + \(3 + t\)\/\@\((3 + t)\)\^2)\) \
+ 6\ \((\(-1\) + \[ExponentialE]\^\(1 - t\/2\))\)\ UnitStep[\(-2\) +
t] + \((6 - 6\ \[ExponentialE]\^\(\(-t\)/2\))\)\ UnitStep[t]\)
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