= Hadamard finite-part integral
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A <Hadamard finite-part integral> removes explicit divergent terms from a truncated <integral> and retains the finite limit. For example, the one-sided finite part of $H(x)/x$ can be defined by
$$
\left\langle\operatorname{Fp}\frac{H(x)}x,\phi\right\rangle=\lim_{\epsilon\downarrow0}\left(\int_\epsilon^\infty\frac{\phi(x)}x\,dx+\phi(0)\log\epsilon\right)
$$
for each <test function> $\phi$. The logarithmic subtraction cancels the divergence at zero. The locally integrable <function> $L(x)=H(x)\log x$, interpreted as zero for $x<0$, has exactly this <distributional derivative>: integrate $-\int_0^\infty\log x\,\phi'(x)\,dx$ by parts with lower cutoff $\epsilon$.
This is distinct from the symmetric <principal-value reciprocal distribution>, which is the derivative of $\log|x|$ on the whole line. More singular reciprocal powers require additional <Taylor polynomial> subtractions. The subtraction and normalization must be specified, because distributions supported at the singular point can distinguish different extensions.
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