Complex integral representation of Dirac delta function?

[pellman]

We all know that $$\frac{1}{2\pi}\int{e^{ik(x-x')}dk=\delta(x-x')$$.

i am working a problem which appears to depend on the statement

$$\int e^{z^*(z-w)}dz^*\propto\delta(z-w)$$

Does anyone know if this is valid?

$$\delta(z-w)$$ is defined in the usual way so that

$$\int{\delta(z-w)f(z)dz}=f(w)$$

This is a physics problem, though, so the domain of integration is not specified and not clear to me.

 

[pellman]

Ha! I just googled "complex delta function" and the first result was this same question posted on the board here over six years ago : https://www.physicsforums.com/showthread.php?t=1253

... by me!

Doesn't anyone know anything about this?

 

[Marin]

Hi, pellman

I just saw your thread and remembered reading a similar article on the net. Here it is, I guess it may be helpful:

http://homepages.physik.uni-muenchen.de/~Winitzki/no_distrib_limit.pdf

 

[pellman]

That paper is a very good jumping off point. Thank you!