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I was wondering if there is a generalization of the integral discussed here to a case like,

\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm \vec{k}_2\vert ^{\nu_3}} \end{equation}

Does some similar Fourier convolution argument again go through?

Is there a generalization to arbitrary number of factors in the denominator?

(..I am using the symbol $\pm$ hoping that like there here too all the $4$ sign combinations probably give the same result..)

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The obvious generalization does not work, since this is no longer a pure convolution integral. In the standard literature, such integrals are handled using Feynman or Schwinger parameters. Have you tried to work those out yourself? – Igor Khavkine May 21 '13 at 12:03
@Igor Khavkine You know of a reference with Feynman parameters with 3 factors as here? Its hard to find such examples in text-books. (..also looking at the 1964 Gelfand-Shillof edition I couldn't locate the identities you used the last time..) – Anirbit May 22 '13 at 9:48
@Anirbit: In standard textbooks, you can find, e.g., eq (6.42) of Peskin & Schroeder and eq (11.A.1) of Weinberg's QFT, v.1. Sorry, I don't have access to an English edition of G&S, but, in my previous answer, I gave an explicit reference for the formulas and the appendix to the Russian edition where they can be found. – Igor Khavkine May 23 '13 at 10:49

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