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Commit 7f878360 authored by Erik Strand's avatar Erik Strand
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Use different variables

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......@@ -21,18 +21,18 @@ $$
Meanwhile, the Fourier transform of $$f$$ is
$$
\hat{f}(\hat{x}, \hat{y})
= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (\hat{x} x + \hat{y} y)} dx dy
\hat{f}(u, v)
= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (u x + v y)} dx dy
$$
Note that the slice along the $$\hat{x}$$ axis in frequency space is described by
Note that the slice along the $$u$$ axis in frequency space is described by
$$
\begin{align*}
\hat{f}(\hat{x}, 0)
&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i \hat{x} x} dx dy \\
&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i \hat{x} x} dx \\
&= \int_\mathbb{R} p(x) e^{-2 \pi i \hat{x} x} dx \\
&= \hat{p}(\hat{x})
\hat{f}(u, 0)
&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i u x} dx dy \\
&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i u x} dx \\
&= \int_\mathbb{R} p(x) e^{-2 \pi i u x} dx \\
&= \hat{p}(u)
\end{align*}
$$
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