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friend
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I'm trying to understand the multiple limit processes involved with the Dirac delta function. Does it matter which process you do first the integral or the delta parameter that approaches zero?
The closest theorem I found that addresses the order of taking limits is the Dominate Convergence Theorem, seen here at:
http://en.wikipedia.org/wiki/Dominated_convergence_theorem
But I'm not sure it applies to the Dirac delta function since there doesn't seem to be a g(x) for which |D(x)|<g(x) for all x, since D(0)=>00.
This doesn't mean, however, that there is not some other proof that integration commutes with the parameter's limiting process for the Dirac delta function, right?
The closest theorem I found that addresses the order of taking limits is the Dominate Convergence Theorem, seen here at:
http://en.wikipedia.org/wiki/Dominated_convergence_theorem
But I'm not sure it applies to the Dirac delta function since there doesn't seem to be a g(x) for which |D(x)|<g(x) for all x, since D(0)=>00.
This doesn't mean, however, that there is not some other proof that integration commutes with the parameter's limiting process for the Dirac delta function, right?