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Regularization of a divergent integral in several variables

  1. Aug 31, 2010 #1
    i've got the following problem

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

    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

    [tex] \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}) [/tex]

    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. jcsd
  3. Aug 31, 2010 #2
    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.
  4. Aug 31, 2010 #3
    however, what would happen if this singularities are OFF the region of the integration , for example

    [tex] \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty}dxdydz \frac{xyz}{xy+yz+zx+2} [/tex]

    has NO poles or singularities on the interval [0,oo) x[0,oo) x[0,oo)
  5. Sep 1, 2010 #4
    One might have some luck with the Laurent expansion in that case.
  6. Sep 2, 2010 #5
    could not one use a change to polar coordinates so the integral is now

    [tex] \int_{0}^{\infty}r^{2}drF(r, \theta , \phi )d\theta \dphi [/tex]

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