Delta Function of Complex Number: How to Define

[zetafunction]

how to define this ??

$$\delta (x-a-ib)$$

here a and b are reals so we are delaing with the delta function of a COMPLEX number

can it be ??

 

[mathman]

The delta function is meaningful only under an integral sign. What is the integral?

 

[zetafunction]

yes of course, but according to this and from the properties of diract delta fnction only for COMPLEX PATH would be the integral non-zero wouldn't it ?

 

[HallsofIvy]

That sentence makes no sense.

The delta function of complex numbers is exactly the same as for real numbers: $\delta(x- a- bi)$ is the distribution such that $\int_B f(x)\delta(x-a-bi)dx= f(a+bi)$ if a+ bi is in the set B, 0 if not.