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Are mathematical manipulations admisible if integral are divergents ?

  1. Sep 5, 2010 #1
    are mathematical manipulations admisible if integral are divergents ??

    are formal manipulations of divergent integrals admisible whenever the integral are divergent [tex] \infty [/tex]

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

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

    [tex] \int_{0}^{\infty}\int_{0}^{\infty}dxdy \frac{(xy)^{2}}{x^{2}+y^{2}+1} [/tex]

    this is an overlapping divergence but if i change to polar coordinates then i should only care about [tex] \int_{0}^{\infty}dr \frac{r^{3}}{r^{2}+1} [/tex] which is just a one dimensional integral.
     
  2. jcsd
  3. Sep 5, 2010 #2

    tom.stoer

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    Science Advisor

    Re: are mathematical manipulations admisible if integral are divergents ??

    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.
     
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