Double Integral with Dirac Delta Function and Changing Limits

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Discussion Overview

The discussion revolves around evaluating a double integral involving a Dirac delta function and the hyperbolic sine function. Participants explore various approaches to handle the integral, particularly focusing on the implications of the delta function and the limits of integration.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the integral and seeks guidance on how to evaluate it, noting the presence of the delta function.
  • Some participants suggest integrating the hyperbolic sine function and express confusion about the role of the delta function in the integral.
  • There are discussions about the conditions under which the delta function is non-zero, particularly regarding the relationship between the parameters a, q, and s.
  • Several participants propose splitting the integral based on the value of q relative to a, indicating that this could clarify the evaluation process.
  • One participant argues that if q, s, and a are all negative, the delta function would always be zero, while another challenges this assertion.
  • There is a suggestion to change the variable in the integral to facilitate evaluation, with some participants expressing skepticism about the correctness of proposed solutions.
  • Another participant emphasizes the importance of correctly identifying the ranges for the parameters involved in the integral.
  • Some participants express frustration with the responses received, indicating a lack of consensus on the best approach to take.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the evaluation of the integral. Multiple competing views and approaches are presented, with ongoing debate about the implications of the delta function and the correct handling of the integral limits.

Contextual Notes

There are unresolved questions regarding the assumptions about the parameters a, q, and s, particularly their ranges and how they affect the evaluation of the integral. The discussion reflects varying interpretations of the delta function's behavior in the context of the given integral.

  • #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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