= Fourier transform of an algebraic cusp
{title2=$\mathcal F(|x|^p)=2\Gamma(p+1)\cos[\pi(p+1)/2]\,|k|^{-p-1}$}
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 $-12|k|^{-4}+2\,9!|k|^{-10}+O(|k|^{-16})$.
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