A Double Integral with Dirac Delta Function and Changing Limits

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The discussion revolves around evaluating a double integral involving a Dirac delta function and the hyperbolic sine function. Participants highlight the challenges posed by the delta function and the variable limits of integration, particularly the need to consider cases where the parameter 'a' falls within the integration bounds. Suggestions include splitting the integral into parts where the delta function is either zero or non-zero, and adjusting limits accordingly. The conversation emphasizes the importance of understanding the behavior of the delta function in relation to the integration variables. Ultimately, the resolution involves careful analysis of the integral limits and the properties of the delta function to simplify the expression.
  • #31
Thank you.
 
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  • #32
Vanadium 50 said:
This has been painful to watch. PeroK has been giving excellent advice.

  • First, split the integral into two pieces, one where the delta function is zero everywhere and one where it is not.
  • Do the inner integral. The first part (above) is zero and the second part (above) sets s = -a. The only q left should be inside the sinh.
  • Set a new variable r = q + a. Set you limits in terms of r.
  • Do the outer (and only remaining) integral. I believe you will have only one a left.
Actually I did that, and I posted the function. Why was that incorrect?
 
  • #33
You can do it pretty easily just be inspecting the integral limits and looking at the interval on which the delta function is nonzero. More specifically, changing the lower bound on the outer integral to ##-a## projects out the integration interval on which the delta function is "satisfied".
 

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