12.2 Fourier Transforms of $U[x]$, $U'[x]$, and $U”[x]$

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The following three plots illustrate the Fourier transforms of the distributional representations of $U(x)$, $U'(x)$, and $U”(x)$ defined in formulas (7), (8), and (9) on page 10 above in blue and the approximations to the Fourier transforms of the Fourier series representations of $U(x)$, $U'(x)$, and $U”(x)$ described on page 10 above in orange. All three plots are from $z=-3$ to $z=3$ and use the same evaluation limit $\epsilon=0.1$. The second plot below corresponds to the evaluation of the real part of the Fourier transform of $U'(x)$, and the first and third plots below correspond to the evaluations of the imaginary parts of the Fourier transforms of $U(x)$ and $U”(x)$ respectively.

$\Im[\mathcal{FT}_x[U(x)](z)]$ from $z=-3$ to $z=3$
$\Re[\mathcal{FT}_x[U'(x)](z)]$ from $z=-3$ to $z=3$
$\Im[\mathcal{FT}_x[U”(x)](z)]$ from $z=-3$ to $z=3$