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I have the following function: $t(x) = e^{-j k_0 d_0}e^{-i (n-1) k_0 \frac{d_0}{2} \cos(2\pi x/\lambda))}$, which can be written in a Fourier series as $t(x) = \sum_q(C_q e^{-i q 2 \pi x/\lambda})$, where $C_q$ are the Fourier coefficients. However, I am relatively new to Fourier series and am really confused about the steps involved in this derivation. Could somebody help me out?

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it is quite simple. You just decompose a function $f(x)$ into the weighted sum of sines and cosines. Did you have a look at : ? – Seyhmus Güngören Sep 23 '12 at 22:06
It would be very advisable you go to the site's FAQ and read there (3rd. paragraph) about how to properly write mathematics here with LaTeX. Your expression for $\,t(x)\,$ looks so absurdly messy that it is very likely many people here don't even try to understand it and leave the question behind...It also be nice if you write $\,i\,$ instead $\,j\,$ for the imaginary unit as this is the usual mathematical symbol for it, unlike what happens sometimes in physics. – DonAntonio Sep 24 '12 at 11:10
I cleaned it up a bit. – John Roberts Sep 24 '12 at 14:37
I'm having no luck with this one. Can somebody help me improve this question? – John Roberts Sep 26 '12 at 13:52

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