Dirac Delta Function: Does it Have a Residue?

[elfmotat]

Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).

 

[pwsnafu]

Does the Dirac delta fuction have a residue? Given the close parallels between the sifting property and Cauchy's integral formula + residue theory, I feel like it should. Unfortunately, I have no idea how they tie together (if they do at all).

This is studied in hyperfunction theory, where you write a generalized function as the "difference" between two analytic functions.

For the record, your intuition is spot on: writing the Dirac as a hyperfunction is a restatement of Cauchy.

 

[elfmotat]

Thank you, that was really helpful.