Fourier Example

Find the Fourier Series of the function:

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

Solution

Here we have

$$\begin{gathered} a_0=\frac{1}{2\pi }(\underset{-\pi }{\overset{0}{\int }}0 \\ dx+\underset{0}{\overset{\pi }{\int }}\pi \\ dx)=\frac{\pi }{2} \end{gathered}$$

$$\begin{gathered} a_n=\underset{0}{\overset{\pi }{\int }}\pi cos(nx) \\ dx=0,n\ge 1, \end{gathered}$$

and

$b_n=\underset{0}{\overset{\pi }{\int }}\pi \sin (nx)dx=\frac{1}{n}(1-cos(x\pi ))=\frac{1}{n}(1-(-1{)}^n)$

We obtain $b_{2n}$ = 0 and

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

Therefore, the Fourier series of f(x) is

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

References:

Fourier Series: Basic Results