Residue of Dirac delta function?

[FredMadison]

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?

 

[stephenkeiths]

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.

 

[h6ss]

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

 

[FredMadison]

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.

 

[blue_raver22]

The Dirac delta function is a mathematical construct that is commonly used in physics and engineering to represent a point mass or point charge. It is not a traditional function in the sense that it does not have a well-defined value at any point, but rather it is defined as a distribution.

In the context of residues, the Dirac delta function does not have a traditional residue as it is not an analytic function. However, it does have a generalized residue defined in terms of its action on test functions. This concept is known as the "principal value" of the Dirac delta function.

To calculate the principal value of the Dirac delta function, we can use the definition of the Dirac delta as a limit of a sequence of functions. This allows us to approximate the Dirac delta function with a sequence of functions that do have well-defined residues, and then take the limit as the sequence approaches the Dirac delta. This concept is similar to how we define the value of a distribution at a point.

In summary, while the Dirac delta function does not have a traditional residue, it does have a well-defined principal value that can be used in calculations involving residues.