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I had an exam question today that stated something along the lines of the following:

"Let $f$ be an even function given by $f(x)=x$ on $[0,\pi]$ and extend $f$ to $\mathbb{R}$ by $2\pi$-periodicity. Find the Fourier series of $f$"

Now for this reason I took $f$ to be $|x|$ on $[-\pi,\pi]$, and at any rate, took my coefficients for $a_n$ and $b_n$ to be $$a_n=\frac{2}{2\pi}\int_0^\pi x\cos(nx)dx$$ for $n\neq 0$, $$a_0=\frac{1}{\sqrt{2\pi}}\int_0^\pi xdx$$ and $b_n=0$ $\forall n\in\mathbb{N}$, with corresponding Fourier series $$|x|\sim\sum_{n=1}^\infty a_n\cos(nx)+b_n\sin(nx).$$

However, I know from a later part of the question that my answer was wrong. Where did I go wrong here? Many thanks in advance for any help that anyone can offer.

Hosein Rahnama
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Dan Burrows
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  • $a_n = \frac{4}{2\pi} \int_0^\pi x \cos(nx) dx$ ? – reuns Jan 12 '16 at 19:29
  • and it is easy to see why $a_0 = \pi/2$ (remember the rest of the series has $0$ mean) : you messed up with the constants. – reuns Jan 12 '16 at 19:34
  • I simply have no idea anymore! I had $\frac{2}{\sqrt{2\pi}}$ as the coefficient to the $a_n$ integral as during the course we used the orthonormal basis ${e_k=\frac{1}{\sqrt{2\pi}}e^{ikx}: k\in\mathbb{Z}}$, I'm not sure why this has been edited. – Dan Burrows Jan 12 '16 at 19:43
  • you know how to recognize an orthogonal matrix ? think to the Fourier coefficients as the result of a matrix (whose lines are orthogonal functions) times your periodic function. and do that check of orthogonality (norm of each vector and scalar product between those) on the Fourier series basis : $\sin\frac{2 \pi n x}{T}$ and $\cos\frac{2 \pi n x}{T}$ or $\exp(\frac{2 i \pi n x}{T})$ – reuns Jan 12 '16 at 20:05
  • ok, thanks for taking the time to help – Dan Burrows Jan 12 '16 at 20:17

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