# Definition of frequency domain coordinates

I am using the Fourier Transform in Optics to perform differentiation with a filter by making use of the relation

$\frac {\partial}{\partial x} f(x)=2\pi i \int^{\infty}_{-\infty} u F(u) \exp (2i\pi ux)\mathrm{d}u$.

In this context, what is the $u$ multiplying $F(u)$ defined as? I have it as a linearly increasing function from 0 to 1, representing a filter of constant gradient function such as $\alpha x+c$, but I don't fully understand how $x$ and $u$ can be related. I have seen something of the form $u=\frac {x}{\lambda D}$, with $\lambda$ being the wavelength and $D$ the distance to the image plane. I cannot see where this has come from or indeed if it is correct.

Thank you in advance.

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You may be interested in my answer here explaining about plane wave components, also known as spatial frequencies. –  ptomato Apr 13 '13 at 16:40