# Tricky integration of DiracDelta function

Posted 8 years ago
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 Hi all,Could you please help me interpret the output of the following lines?I understand the first result, because the integration starts exactly at -alpha, and so we get the Heavisidetheta function evaluated at zero. However, the other two don't make much sense to me, especially the last one, which is equal to zero for any function of s multiplied by the delta function!!Thanks, Pedro
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Posted 8 years ago
 Makes sense. Thank you very much!!
Posted 8 years ago
 Mathematica sometimes cannot handle manipulations involving the delta function. If you put in by hand the identity delta(x/S)=Abs(S)delta{x), then everything works out reasonably. (I am writing sigma->S and alpha->a, so this shows up more cleanly in the code.) In[40]:= Assuming[a > 0, Integrate[DiracDelta[s + a], {s, -a, 1}]] Out[40]= HeavisideTheta[0] In[41]:= Assuming[a > 0, Integrate[Abs[S] DiracDelta[s + a], {s, -a, 1}]] Out[41]= Abs[S] HeavisideTheta[0] In[42]:= Assuming[a > 0, Integrate[Abs[S] DiracDelta[s + a] f[s], {s, -a, 1}]] Out[42]= Abs[S] f[-a] HeavisideTheta[0]