Are mathematical manipulations admisible if integral are divergents ?

[zetafunction]

are mathematical manipulations admisible if integral are divergents ??

are formal manipulations of divergent integrals admisible whenever the integral are divergent $$\infty$$

i mean if i have a 4-dimensional integral $$\int_{R^{4}}dxdydzdt F(x,y,z,t)$$

why can we make a change of variable to polar coordinates ?? , or for example if we have an UV divergent integral $$\int_{0}^{\infty}dxx^{4}$$ by means of a change of variable $$x= 1/y$$ this integral is IR divergent $$\int_{0}^{\infty}dyy^{-6}$$ or if i have the divergent integral

$$\int_{0}^{\infty}\int_{0}^{\infty}dxdy \frac{(xy)^{2}}{x^{2}+y^{2}+1}$$

this is an overlapping divergence but if i change to polar coordinates then i should only care about $$\int_{0}^{\infty}dr \frac{r^{3}}{r^{2}+1}$$ which is just a one dimensional integral.

 

[tom.stoer]

I think that all manipulations have to be backed up by some cutoff procedure.

So given a divergent integral this is just mathematical nonsense. Introducing a cutoff, doing some manipulations and concluding that the two well-defined integrals are identical is fine. In this sense it may be reasonable to conclude that two divergent integrals are "identical", namely because their finite counterparts are related.