 Hadamard finite-part integral

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 byfor each test function $\varphi$. 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 $-{\int }_0^{\infty }\log x{\varphi }^{\prime }(x)dx$ by parts with lower cutoff $ϵ$.

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.