Example: Periodic parabolic function

3.11.4 Example: Periodic parabolic function

$f(x)=x(2\pi-x),\;\;0\le x\le2\pi\;\;$ periodic continuation

\begin{eqnarray*} \frac{a_0}{2}&=&\frac{1}{2\pi}\int\limits_0^{2\pi}(2\pi x-x^2)dx=\frac{1}{2\pi}\left[\pi x^2-\frac{1}{3}x^3\right]_0^{2\pi}=\frac{2}{3}\pi^2\\ a_k&=&\frac{1}{\pi}\int\limits_0^{2\pi}2\pi x\cos kx\;dx-\frac{1}{\pi}\int\limits_0^{2\pi}x^2\cos kx\;dx\\ &=&\frac{1}{\pi}\left[2\pi\left(\frac{\cos kx}{k^2}+\frac{x\sin kx}{k}\right)-\left(\frac{2x}{k^2}\cos kx+\left(\frac{x^2}{k}-\frac{2}{k^3}\right)\sin kx\right)\right]_0^{2\pi}\\ \frac{1}{\pi}\left(-\frac{4\pi}{k^2}\right)&=&-\frac{4}{k^2}\\ \mbox{Thus:}\quad f(x)&=&\frac{2}{3}\pi^2-4\sum_{k=1}^\infty \frac{\cos kx}{k^2} \quad \Rightarrow \quad \mbox{Due to $1/k^2$ fast converging series} \end{eqnarray*}

The following animation shows the Fourier-approximation for k up to 6.

With frame

Fourier series