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I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will include the link).

I encounter an integral at some point, which I am having a lot of difficulty to prove: the integral is this $$\int\frac{d^3\vec{k}}{2k^0}\sin\bigg[k\cdot\bigg(\frac{q}{q^0}\tau-\frac{p}{p^0}s\bigg)\bigg]=2\pi^2\delta\bigg[\bigg(\frac{q}{q^0}\tau-\frac{p}{p^0}s\bigg)^2\bigg]\varepsilon(\tau-s)$$ where $\varepsilon(x)\equiv{\rm sgn}(x)$. So, how do I prove that this integral holds? If so, how do I perform an integral over $s$ at some later stage since the delta function has a power of two in its argument i.e. what is the integral $\int dx\ \delta[(x-a)^2]f(x)$?

Any help will be appreciated

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Not a complete answer, but if $f(x)$ is zero at the points $x_n$ we have $$ \delta(f(x)) = \sum_{x_n} \frac 1{|f'(x_n)|} \delta(x-x_n). $$

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  • $\begingroup$ Yes it seems that it works... I wasn't applying the formula correcltly. Now, however, it remains to prove that the integral holds! Thanks though $\endgroup$
    – schris38
    Apr 30, 2022 at 7:10

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