How do I show that if $f(x,y)$ is separable into a product of a function of $x$ and a function of $y$, its Fourier transform $F(u,v)$ is also separable into a function of $u$ and a function of $v$?
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$\int\int f(x,y)\ e^{i 2\pi (vx+uy)} \ dx dy = \int\int f(x)f(y)\ e^{i 2\pi vx} e^{i 2\pi ux}\ dx dy = \int f(x)e^{i 2\pi vx} dx \int f(y)e^{i 2\pi ux}\ dy = F(u)F(v)$
George
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1I highly suggest not using $j$ for the imaginary unit on Math Stack Exchange to avoid confusion. $j$ is primarily used by engineers, whereas in math courses, $i$ is preferred. – Cameron Williams Oct 02 '14 at 23:52
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Furthermore: for OP, the interchanging of the integrals can be justified by Fubini's theorem. – Cameron Williams Oct 02 '14 at 23:56
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You are very accurate @CameronWilliams – George Oct 03 '14 at 00:02