Regularization of a divergent integral in several variables

[zetafunction]

i've got the following problem

let be the integral $$\int_{R^{3}} dxdydz \frac{R(x,y,z)}{Q(x,y,z)}$$

here R(x,y,z) and Q(x,y,z) are Polynomials on several variable.

Let us suppose this integral is divergent, in order to regularize it i have thought about substracting several terms so

$$\int_{R^{3}} dxdydz( \frac{R(x,y,z)}{Q(x,y,z)}-\sum_{i,j,k} C_{i,j,k}(x+a)^{i}(y+b)^{j}(z+c)^{k})$$

is convergent, here i,j,k can run from -1 to up a certain finite integer

for the one dimensional case i know how to do it but i have not any idea to generalize to several variables

 

[Eynstone]

Meromorphic functions in two or more variables don't have isolated singularities. Hence we can't use the form you've suggested & use functions in x,y,z, Further, non-homogeneous rational functions don't split into partial fractions easily.

 

[zetafunction]

Meromorphic functions in two or more variables don't have isolated singularities. Hence we can't use the form you've suggested & use functions in x,y,z, Further, non-homogeneous rational functions don't split into partial fractions easily.

however, what would happen if this singularities are OFF the region of the integration , for example

$$\int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty}dxdydz \frac{xyz}{xy+yz+zx+2}$$

has NO poles or singularities on the interval [0,oo) x[0,oo) x[0,oo)

 

[Eynstone]

however, what would happen if this singularities are OFF the region of the integration

One might have some luck with the Laurent expansion in that case.

 

[zetafunction]

could not one use a change to polar coordinates so the integral is now

$$\int_{0}^{\infty}r^{2}drF(r, \theta , \phi )d\theta \dphi$$

integration over the angles will make the integral to be one dimensional