Example: Absolute of sin function

3.11.3 Example: Absolute of sin function

$f(x)=|\sin x|\quad$ periodic in $\pi$

even function $\Rightarrow b_k\equiv$0 \begin{eqnarray*} \frac{a_0}{2}&=&\frac{1}{\pi}\int\limits_0^\pi\sin x\;dx=\frac{1}{\pi}\left[-\cos x\right]_0^\pi=\frac{2}{\pi}\\ a_k&=&\frac{2}{\pi}\int\limits_0^\pi\sin x\cos2kx\;dx=\frac{2}{\pi}\left[-\frac{\cos(1+2k)x}{2(1+2k)}-\frac{\cos(1-2k)x}{2(1-2k)}\right]_0^\pi\\ &=&\frac{2}{\pi}\left[\frac{1}{1+2k}+\frac{1}{1-2k}\right]=-\frac{4}{\pi}\frac{1}{4k^2-1}\\\\ \mbox{Thus:}\quad f(x)&=&\frac{2}{\pi}-\frac{4}{\pi}\sum_{k=1}^\infty\frac{1}{4k^2-1}\cos2kx \quad \Rightarrow \quad \mbox{Due to $1/(4k^2-1)$ fast converging series} \end{eqnarray*}

The following animation shows the Fourier-approximation for $f(x)=x(2\pi-x),\;\;0\le x\le2\pi\;\;$ up to 6.

With frame

Fourier series