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I've been working with this equation where the unknown factor is the function $f$ that can be complex:

$$1 = f(\vec{x})\int_{\mathbb{R}^3}\ d^3y\ \frac{f(\vec{y})}{|\vec{x} - \vec{y}|^4}$$

Is there any way to solve this equation without using the trial-error method, i.e., without testing different forms for $f$? Anyway, can you see any solution?

Thanks in advance!

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    $\begingroup$ Is there not a problem with the integral when $y=x$? $\endgroup$ – user121049 Nov 1 '18 at 19:06
  • $\begingroup$ I don't follow you, what do you want to say? $\endgroup$ – Vicky Nov 1 '18 at 19:35
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    $\begingroup$ Forget solving for $f$, I don't see how you choose a $f$ so that the integral is finite for any $x$, but maybe that's my lack of vision. $\endgroup$ – user121049 Nov 1 '18 at 20:02
  • $\begingroup$ How do we deal with the singularity at $y=x$? $\endgroup$ – Yuriy S Nov 1 '18 at 20:07
  • $\begingroup$ Well, from the point of view of the integral $x$ is just a parameter, so for integration I think you can suppose $x \neq y$ $\endgroup$ – Vicky Nov 1 '18 at 20:28

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