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Lately I came across the following identity:

$lim_{\eta\rightarrow0}\lim_{\delta\rightarrow0}2\eta\frac{1}{\omega-z_1-i\eta}\frac{1}{\omega-z_2+i\eta}\frac{1}{z_3-\epsilon-i\delta}\frac{1}{z_2+z_3-z_1-\epsilon+i\delta}=(2\pi)^3\delta(z_1-\omega)\delta(z_2-\omega)\delta(z_3-\epsilon),$

where $\omega,\epsilon,\eta,\delta\in \mathbb{R}$ and $z_i\in\mathbb{R}$.

I can test it by applying the left and the right sides of the identity to a test analytic function, such as $e^{iaz_1}e^{biz_2}e^{icz_3}$ for $a,b,c\in\mathbb{R}$.

My question is two-fold:

i) Do you happen to know if such identity has a name, a la Sokhotsky formula

ii) What would be a general strategy starting from a specific case of analytic test function to prove the identity for more general test functions?

Thank you in advance!

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  • $\begingroup$ The question inevitably and naturally has a rather involved background, namely, the Nonlinear Algebraic Theory of Generalized Functions, listed under 46F30 by the American Mathematical Society in their Mathematics Subject Classification. And the main related point is that the multiplication of generalized functions, and thus in particular, of delta-functions, does INEVITABLY branch into infinitely many different possible multiplications, and does so above certain levels of singularities. $\endgroup$
    – Elemer E Rosinger
    Commented Sep 8, 2014 at 9:39
  • $\begingroup$ In other words, there are infinitely many different ways of multiplication of such singular generalized functions. Consequently, the question asked does NOT make sense, unless first one specifies or chooses one of those infinitely many different possible multiplications of generalized functions. Amusingly, as it happens, this inevitable and natural branching of multiplication is not quite well understood even among quite a number of specialists in the field. And it is not understood either by such a remarkable scholar like Roger Penrose. $\endgroup$
    – Elemer E Rosinger
    Commented Sep 8, 2014 at 9:39
  • $\begingroup$ Consequently, time an again, questions like the present one arise, and answers are sought, without giving any additional sine-qua-non information, thus making it impossible to give a correct answer. $\endgroup$
    – Elemer E Rosinger
    Commented Sep 8, 2014 at 9:39
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    $\begingroup$ @ElemerERosinger : The complications you mention indeed arise when you consider generalized functions of the same argument, e.g $\delta(z_1-a)\delta(z_1-b)$. Here we have a simpler situation where the rhs can be considered as a function of 3 independent variables $z_1$, $z_2$, $z_3$. $\endgroup$
    – yarchik
    Commented Sep 8, 2014 at 10:15

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