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This notebook's Convolve won't symbolically evaluate this one dimensional convolution in tau:

Convolve[DiracDelta[Sqrt[1+tau^2]],DiracDelta[tau],tau,z]

Is this because the variable tau appears inside Sqrt? If so, how can one get Mathematica to perform this symbolic convolution?

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  • $\begingroup$ As far as I know the product of DiracDeltas isn't defined. $\endgroup$ Commented Mar 27, 2022 at 15:38
  • $\begingroup$ @UlrichNeumann I don't argee. Mathematica evaluates the following simpler function Convolve[DiracDelta[tau - r0], DiracDelta[tau], tau, z] $\endgroup$
    – mikado
    Commented Mar 27, 2022 at 16:14
  • $\begingroup$ I didn't know how to make DiracDelta plural in a title such that it wouldn't be confused with me asserting there was such a thing as Wolfram language function DiracDeltas. $\endgroup$ Commented Mar 27, 2022 at 17:59
  • $\begingroup$ @mikado Thanks for your comment. You're right, via Fouriertransform the convolution is defined! $\endgroup$ Commented Mar 28, 2022 at 6:02

1 Answer 1

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To evaluate the convolution you need the transformation rule

enter image description here

( see wikipedia )

xi = Solve[Sqrt[rho^2 + tau^2] - r0 == 0, tau]
gsi = D[Sqrt[rho^2 + tau^2] - r0, tau] /. xi
res = (Flatten[xi] /. Rule -> Subtract)
dirac = Total@MapThread[DiracDelta[#1]/ Abs[#2] &, {res, gsi}]

convolution

Convolve[dirac, DiracDelta[tau], tau, z]
(*DiracDelta[Sqrt[r0^2 - rho^2] - z]/Abs[Sqrt[r0^2 - rho^2]/Sqrt[r0^2]] + 
DiracDelta[Sqrt[r0^2 - rho^2] + z]/Abs[Sqrt[r0^2 - rho^2]/Sqrt[r0^2]] *)
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  • $\begingroup$ You should add a caveat to your answer that it doesn't work in the case of multiple dimensions. This caveat is needed because the example given uses a fairly common cylindrical coordinate form. Indeed, if I'd correctly understood the problem I was really trying to solve, I would have stated in the title that it was in cylindrical coordinates, as I did in this question: math.stackexchange.com/questions/4414678/… $\endgroup$ Commented Mar 29, 2022 at 13:09
  • $\begingroup$ Don't understand, what you mean: Here we have a onedimensional integration along tau with scalar argument inside DiracDelta I think. $\endgroup$ Commented Mar 29, 2022 at 16:20
  • $\begingroup$ I changed the question to make your answer not mislead those of us who have more than 1 dimension to deal with. So your answer is ok without the caveat now. Although You might consider changing the formula in your answer to match: Sqrt[1+tau^2] == 0 $\endgroup$ Commented Mar 30, 2022 at 14:42

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