How Does Using Dirac in a Double Integral Simplify the Integration Process?

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Discussion Overview

The discussion centers on the use of the Dirac delta function in the context of a double integral involving sine functions. Participants explore how the Dirac function may simplify the integration process, particularly when considering limits of integration.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant poses a question about whether the double integral involving the Dirac delta function simplifies to a single integral.
  • Another participant notes that the absence of limits of integration affects the interpretation of the integral and provides a general condition for the Dirac delta function.
  • A later reply clarifies that the limits of integration are from negative infinity to positive infinity.
  • One participant expresses agreement with the simplification but raises a caution about potential issues with non-converging integrals.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the simplification process, as there are differing views on the implications of the limits of integration and the behavior of the integral.

Contextual Notes

The discussion highlights the importance of specifying limits of integration when using the Dirac delta function, as well as the potential for non-convergence in certain cases.

ooga59
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Hi,

I have a question about using the dirac function in a double integral.

Lets say you have the double integral over the two values x1 and x2:

int( int( sin(x1) * dirac(x1-x2) * sin(x2) ))

Does this just simplify to a single integral:

int( (sin(x1))^2 )



thanks!
 
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Well, that depends. You did not write any limits of integration.
[tex]\int_a^b \delta(x- x1)f(x)dx[/tex]
is equal to f(x1) IF [itex]a\le x1\le b[/itex] and 0 otherwise.
 
The limits would be from -inf to inf. Thanks.
 
AFAIK the answer is yes. But of course that non converging integral might produce some surprises.
 

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