Fourier Example

Find the Fourier Series of the function:

$f(x)={\begin{cases}0,&-\pi <x<0\\\pi ,&0<x<\pi \\\end{cases}}$

$b_{n}={\frac {1}{2\pi }}\int _{0}^{\pi }\pi \sin(nx)\,dx,={\frac {1}{n}}(1-cos(x\pi ))={\frac {1}{n}}(1-(-1)^{n})$

We obtain b_2n = 0 and

$b_{2n+1}={\frac {2}{2n+1}}$

Therefore, the Fourier series of f(x) is

$f(x)={\frac {\pi }{2}}+2(sin(x)+{\frac {sin(3x)}{3}}+{\frac {sin(5x)}{5}}+...)$