Iterative integration in several variables

[zetafunction]

as a physicist , there are some integrals in several variables that are DIVERGENT

in order to regularize them i argue that for any integral in several variables

$$\int_{V} f(x,y,z)dV$$

you can always perform an interative integration (you integrate in variable 'x' for example keeping the others variables as constant and then you regularize the result in each variable by using the Hadamard's finite part integral definition)

but is this valid ?? .. even for a divergent integral (let us suppose we introduce a cut-off so the integral is rendered finite and then we take the cut-off limit --> oo ) is this valid we can perform an integral in several variables by doing an interation of one dimensional integrals ??

let us suppose i introduce the regulators $$((x+a)((y+b)(z+c))^{-s}$$

for a big 's' so the integral is convergent and then i take the limit (by analytic continuation ) to s -->0

 

[fresh_42]

but is this valid ??

Divergence in mathematics cannot be renormalized away. So whether a manipulation is "valid" depends largely on the background and purpose.