Can Distribution Theory Extend to Complex Integrations?

[zetafunction]

given a differentiable function g(x) we know that in many cases if we define $$g(nx)$$ for n--> oo , then we have no longer a function but a DISTRIBUTION

example $$\delta (x) = \frac{sin(Nx)}{x}$$ as n--->oo

could the same be applied in distribution theory ? for example

$$T(n,x)= \sum_{i=0}^{n}\delta (x-i)$$

and for the complex integration could we consider

$$\int_{C}dsF(s)x^{s}/s$$ ,

here F(s) is a test function in complex plane and $$x^{s}/s$$ is a distribution on parameter 's' , and x is a real constant.

 

[Aaron Barnard]

Yes, the same principles can be applied to distribution theory. In the example given above, the function T(n,x) is a Dirac comb, which is a type of distribution. In the second example, the integral is a Mellin transform, which is also a type of distribution. Both of these are special cases of the more general concept of a distribution.