Regularization of a divergent integral in several variables

1. Aug 31, 2010

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

2. Aug 31, 2010

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.

3. Aug 31, 2010

zetafunction

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)

4. Sep 1, 2010

Eynstone

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

5. Sep 2, 2010

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