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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?

Integrating Dirac Delta Function

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

POSTED BY: Pedro Gardete
2 Replies
Posted 8 years ago

Makes sense. Thank you very much!!

POSTED BY: Pedro Gardete

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]
POSTED BY: S M Blinder
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