Delta derivatives from polynomial oscillatory amplitudes
= Delta derivatives from polynomial oscillatory amplitudes
{title2=$\int e^{is\theta}\theta^j\,d\theta=2\pi i^{-j}\delta^{(j)}(s)$}
With inverse <Fourier transform> normalization $(2\pi)^{-1}$, $\int e^{is\theta}\theta^j\,d\theta=2\pi i^{-j}\delta^{(j)}(s)$ as an <oscillatory integral>. The factor $i^{-j}$ and the sign convention $\langle\delta',g\rangle=-g'(0)$ are essential. For instance, amplitude $-ix_2\theta/(2\pi)$ with phase $x_1\theta$ gives $-x_2\delta'(x_1)$.