Hadamard finite-part reciprocal-power distribution (${\Lambda }_m=\mathrm{Fp}(x^{-m})$)

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.