Zeta function for complex argument

[zetafunction]

can we really give a definition of $$\delta (x-a-ib)$$ a,b real and 'i' means the square root of -1

if i try it in the sense of generalized function for any x a and b i get the result oo unless b is zero

 

[HallsofIvy]

can we really give a definition of $$\delta (x-a-ib)$$ a,b real and 'i' means the square root of -1

if i try it in the sense of generalized function for any x a and b i get the result oo unless b is zero

No, you don't. A generalized function (distribution) does not HAVE a value at a specific point. A generalized function is a "functional" that assigns a number to every function. We can think of the functions as a subset of the generalized functions by saying that the function f(x) is the functional that to any function g(x) assigns the number $\int_A f(x)g(x)dx$ where "A" is some given set we use to define our generalized functions.

The standard delta function, $\delta(x)$ is the functional that assigns f(0) to every function f. The "shifted" delta function, $\delta(x- a)$ is the functional that assigns f(a) to every function f. For a complex number, a+ bi, $\delta(x-a-bi)$ is the functional that assigns the value f(a+bi) to every function f. Of course, for that to make sense we must be talking about functions of complex numbers, not functions of real numbers only.