= Principal-value reciprocal distribution
{title2=$\operatorname{pv}(1/x)$}
The <Cauchy principal value>
$$
\left\langle\operatorname{pv}\frac1x,\varphi\right\rangle
=\lim_{\varepsilon\downarrow0}\int_{|x|>\varepsilon}\frac{\varphi(x)}x\,dx
$$
defines a <distribution>. Near zero the odd constant contribution cancels, and the remaining numerator is $O(x)$. It satisfies $x\operatorname{pv}(1/x)=1$. Hence all solutions of $xv=1$ are $v=\operatorname{pv}(1/x)+c\delta_0$ by the <kernel of multiplication by a coordinate>.
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