Graduate Double Integral with Dirac Delta Function and Changing Limits

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SUMMARY

The discussion revolves around evaluating the double integral involving the Dirac delta function: \int_{-1}^{0}\int_{-1}^{q}\delta(s+a)\sinh[k(q-s)]dsdq, where -1. Participants emphasize the importance of understanding the behavior of the delta function in relation to the integration limits, particularly when q is compared to a. Key strategies include splitting the integral based on the conditions of q and a and recognizing that the delta function is zero when its argument is not satisfied. The final evaluation leads to a simplified expression involving hyperbolic functions.

PREREQUISITES
  • Understanding of double integrals and their properties
  • Familiarity with the Dirac delta function and its implications in integration
  • Knowledge of hyperbolic functions, specifically sinh and cosh
  • Ability to manipulate integration limits based on variable conditions
NEXT STEPS
  • Study the properties of the Dirac delta function in integrals
  • Learn techniques for evaluating double integrals with variable limits
  • Explore hyperbolic functions and their applications in integrals
  • Investigate symbolic algebra tools for complex integral evaluations
USEFUL FOR

Mathematicians, physicists, and engineering students who are working with advanced calculus, particularly in contexts involving integrals with delta functions and hyperbolic functions.

  • #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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