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Integral of DiracDelta

Posted 7 months ago
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Hello

Common sense suggest that the following integral

Integrate[DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], 0, \[Pi]}]

should equal one. But Mathematica returns a 0. ┬┐Is this correct?

Thank you

Carlos

3 Replies
In[1]:= Integrate[
 DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], 0, \[Pi]}]

Out[1]= 0

In[2]:= Integrate[
 DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], -1/10, \[Pi]}]

Out[2]= 1
In[27]:= Integrate[
     DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], Pi/10, Pi}]

    Out[27]= 1

and

In[37]:= Integrate[
 DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], -Pi/10, +Pi/2}]

Out[37]= 1 - HeavisideTheta[0]

This integral is rather usual in Physics, for spherical coordinates.

Integrate[DiracDelta[Cos[\[Theta]]] Sin[\[Theta]], {\[Theta], #, \[Pi]}] & /@ {0, 0.}
(*  Out:   {0,1.}  *}

Strange indeed!

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