In Sato's theory, the following formal delta function is defined:


$\delta(\lambda,z)=\frac{1}{\lambda}\sum\_{n=-\infty}^\infty(\frac{z}{\lambda})^n=\frac{1}{z}\frac{1}{1-\lambda/z}+\frac{1}{\lambda}\frac{1}{1-z/\lambda}$


Given a function 
$f(z)=\sum a\_iz^i$,


$f(\lambda)\delta(\lambda,z)=f(z)\delta(\lambda,z)$.


I want to know the properties as many as possible. Or useful references are welcome to be provided. Thanks!



Who can tell me the properties for the delta function in Sato's theory?
The formal delta function obeys the usual properties that the Dirac delta function does, but relative to the pairing defined by the residue. For instance,
$$ \mathrm{Res}\_z f(z)\delta(z,w) = f(w)$$
for any formal distribution $f(z)$.


This and more can be found in Kac's [*Vertex algebras for beginners*](http://books.google.com/books?id=PIhm9-37IlUC), particularly Proposition 2.1 in §2.1.


