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.

Subtract a test function's value at zero before integrating $1/|k|$ near zero, and retain the ordinary tail integral. This defines an even tempered distribution. Changing the subtraction cutoff adds a multiple of the Dirac delta distribution. Multiplication by $k$ gives $\mathrm{sgn}k$.

For $m\ge 2$, the symmetric Hadamard finite-part integral defining ${\Lambda }_m$ subtracts the Taylor polynomial through degree $m-2$ from a test function before division by $x^m$. It has finite order of a distribution and satisfiesThe lower-order subtraction terms are integrable at infinity, while the remaining odd $1/x$ singularity cancels under symmetric truncation.