= Hadamard finite-part reciprocal-power distribution
{c}
{title2=$\Lambda_m=\operatorname{Fp}(x^{-m})$}
For $m\geq2$, 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 satisfies
$$
\Lambda_m=\frac{(-1)^{m-1}}{(m-1)!}\left(\frac d{dx}\right)^m\log|x|.
$$
The lower-order subtraction terms are integrable at infinity, while the remaining odd $1/x$ singularity cancels under symmetric truncation.
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