Distributional derivative of the logarithmic modulus
= Distributional derivative of the logarithmic modulus
{title2=$\frac d{dx}\log|x|=\operatorname{pv}(1/x)$}
The <locally integrable function> $\log|x|$ has <distributional derivative> equal to the <principal-value reciprocal distribution>. Symmetric <integration by parts> makes the boundary term $\log\varepsilon[\varphi(\varepsilon)-\varphi(-\varepsilon)]$ tend to zero.