Delta derivatives from polynomial oscillatory amplitudes ($\int e^{is\theta }{\theta }^{j}d\theta =2\pi i^{-j}{\delta }^{(j)}(s)$)

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 $⟨{\delta }^{\prime },g⟩=-g^{\prime }(0)$ are essential. For instance, amplitude $-i{x}_2\theta /(2\pi )$ with phase $x_1\theta$ gives $-x_2{\delta }^{\prime }(x_1)$.