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I want to solve the following:

Given the $2\pi$ periodic function $f$:

$$ f(x) = \begin{cases} 2\pi & for \;\; 0 < x < K \\ 0 & for \;\; K < x<2\pi \end{cases} $$

Where K is a constant between $0$ and $2\pi$.

Find the complex Fourier Series of $f$.

I am new to these kinds of problem and I need an informal, step-by-step explanation of how one finds the series. Thanks so much for helping me out!

knuterr
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    Just compute $c_n=(2\pi )^{-1}\int _0 ^K 2\pi e^{-inx},dx $ for any $n\in \mathbb{Z}$. The answer is then $\sum _{n=-\infty }^\infty c_n e^{inx}$. – flavio Nov 06 '13 at 10:15

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A small $caveat$: the function $f$ is not periodic, as it is defined only on $[0,2\pi]$. You need to introduce at first a periodic extension $\tilde{f}$, and then compute the Fourier coefficients of $\tilde{f}$. In the OP the period $T=2\pi$ is considered: you can draw the $2\pi$-periodic extension $\tilde{f}$ of $f$ quite easily.

The definition of Fourier coefficients (contained, for example in here) allows you to use the formulae in jiku's comment, where the period $T=2\pi$ is explicitly chosen.

Avitus
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