Residue of Dirac delta function?

  • #1

Main Question or Discussion Point

Does the Dirac delta have a residue? It seems like it might, but I don't know how to attack it, since I really know very little about distributions. For example, the Dirac delta does not have a Laurent-expansion, so how would you define its residue?
 

Answers and Replies

  • #2
Perhaps you could try using something that approaches a delta function like [itex]\frac{1}{\pi} \frac{sin(\lambda x)}{x} [/itex] as [itex] \lambda \to \infty [/itex] or maybe [itex] \sqrt{\frac{\alpha}{\pi}}e^{-\alpha x^2} [/itex] as [itex] \alpha \to \infty [/itex]

Perhaps you could find the residue w.r.t. [itex] \lambda [/itex] and take the limit. Not sure, just something to maybe try.
 
  • #3
77
9
Have you tried extending the Dirac delta to the complex plane and then examining its poles?
 
  • #4
Thanks for your replies, I have not had time yet to look into this more. I still think it's an interesting question, but my original motivation for it has disappeared, since I realized that a contour integral around a delta-function singularity will give 0, by simply looking at the definition of the complex integral - the integrand is in this case zero everywhere on the contour. I got fooled by thinking that since there is a "spike" inside the contour that would mean that I would have to look at the residues. Since delta(z) is not analytic, nothing can be inferred about its residue from the contour integral.
 

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