Fourier Coefficients given a periodic function

Question

I have been given a function $f\in C_{st}$ which in $[0,\pi]$ is defined as: $$ f(x)=\frac{x}{4}(2\pi-x)$$ for $0\leq x\leq \pi$. Which is an even function with period $2\pi$. Let $c_n$ be the Fourier coefficients regarding the usual orthonormal system $\{e_n|n\in\mathbb{Z}\}$ for $(C_{st},(\cdot,\cdot)).$ I have to find the Fourier coeffiecients, however, the answer i get for $c_0$ is not right. $$ c_0=(f,e_0)=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)e^{-i\cdot0\cdot x}dx=\frac{1}{2\pi}\int_0^\pi (\frac{x}{4}(2\pi-x))dx$$ Calculating this integral gives me that $c_0=\frac{\pi^2}{12}$. However, i am told that the answer should be $c_0=\frac{\pi^2}{6}$. What am i doing wrong?

Answer 1

You've been given a function which is defined on $ [0, \pi] $, but you're computing its Fourier coefficients by viewing it as a function on $ [-\pi, \pi] $ that vanishes on $ [-\pi, 0] $. These are not equivalent procedures. If you do a Fourier series expansion on the interval $ [0, \pi] $ instead, you'll get the right coefficient out of it.