# Residue of Dirac delta function?

## 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?

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

Perhaps you could find the residue w.r.t. $\lambda$ and take the limit. Not sure, just something to maybe try.

Have you tried extending the Dirac delta to the complex plane and then examining its poles?

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.