Hadamard finite-part integral

A Hadamard finite-part integral subtracts the divergent terms of a truncated integral and retains its finite limit. It defines distributions such as the extension of $1/x$ obtained by differentiating $\log x_{+}$.

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.