Finite-part inverse-absolute-value distribution ($\mathrm{Pf}(1/|k|)$)

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 $\mathrm{sgn}k$.