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A Logarithmic divergence of an integral

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  1. Feb 19, 2017 #1
    I would like to prove that the following integral is logarithmically divergent.

    $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$

    This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##.

    However, it is the highest power of ##k## in the denominator that is ##8##. There are other terms in ##k## in the denominator of the form ##k^7##, ##k^6##, etc.

    I was wanting a more formal proof that the integral is logarithmically divergent.
     
  2. jcsd
  3. Feb 22, 2017 #2
    How does this sound?

    Your denominator is a polynomial in k; so it can be written a product of terms like (k - zi) where the zi are zeros of the polynomial. If you break this product into partial fractions, you'll automatically get a log on integration.
     
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