 Fourier transform of an algebraic cusp

For $p>-1$ this formula is interpreted away from $k=0$ as a tempered distribution or by exponential regularization. It also supplies local high-frequency cusp terms after multiplying by a smooth cutoff. Even nonnegative integer $p$ give smooth polynomials and only contact terms at $k=0$, so their localized forms have no algebraic tail. For example, the cusp expansion of $(1+|x{|}^3{)}^{-1}$ gives $-12|k{|}^{-4}+29!|k{|}^{-10}+O(|k{|}^{-16})$.