Does the Dirac Delta function have a residue? I feel like it should, considering the close parallels between delta function integral identities and theorems in complex analysis, namely the Cauchy integral formula and residue theory. I'm unsure exactly how they tie together (if they do).

Edited January 28, 201313 yr by elfmotat

I am not sure either, but I think you would need to extend the Dirac delta to the complex plane and then examine the poles. This sounds like something someone will have examined for sure.

I am not sure either, but I think you would need to extend the Dirac delta to the complex plane and then examine the poles. This sounds like something someone will have examined for sure.

The Dirac delta already makes sense on the complex plane - however, the idea of residues would need to be extended to some subset of the set of distributions, rather than only dealing with (ratios of) functions.

 

One property that it should have: the residue of any derivative should be 0, as can be seen by examining the Laurent series or using the residue theorem.

=Uncool-

Thanks for the input guys. I asked this question on another site as well, and I was referred here: http://en.m.wikipedia.org/wiki/Hyperfunction . So it seems I wasn't too far off.