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Why no change of variable to polar coordinates inside multi-loop integral ?

  1. Jun 12, 2010 #1
    why no change of variable to polar coordinates inside multi-loop integral ??

    given a mul,ti-loop integral

    [tex] \int d^{4}k_{1} \int d^{4}k_{2}.................\int d^{4}k_{n}f(k_{1} , k_{2},.....,k_{n}) [/tex]

    which can be considered a 4n integral for integer n , my question is why can just this be evaluated by using a change of variable to 4n- polar coordinates ?

    one we have made a change of variable and calculated the Jacobian, and integrated over ALL the angular variables we just have to make an integral

    [tex] \int_{0}^{\infty}drg(r)r^{4n-1} [/tex] which is just easier to handle
     
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  3. Jun 12, 2010 #2

    mathman

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    Re: why no change of variable to polar coordinates inside multi-loop integral ??

    I don't what specific integral you have in mind, but it depends very much on the form of f as it depends on the k's. You seem to imply that it can be represented as a function g of one variable. This may be true for some particular f, but it certainly is not true in general.
     
  4. Jun 13, 2010 #3
    Re: why no change of variable to polar coordinates inside multi-loop integral ??

    for example

    [tex] \iint dx dy \frac{x^{3}}{1+xy} [/tex] its divergent if taking the limits (0,oo)

    making a change of variable to polar coordinates one gets

    [tex] \int du \int_{0}^{\infty}dr\frac{r^{4}cos^{3}(u)}{1+(1/2)r^{2}sin(2u)} [/tex]

    integrating over the angular variable 'u' you have now a simple one dimensional integral
     
  5. Jun 13, 2010 #4

    mathman

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    Re: why no change of variable to polar coordinates inside multi-loop integral ??

    In general if you have an m dimensional integral and integrate over m-1 dimensions, you will have a one dimensional integral. In your general case (4n) I am not sure what you mean by polar coordinates.

    This question belongs in the mathematics forum. There isn't apparent connection with Beyond the Standard Model (physics).
     
    Last edited: Jun 13, 2010
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