Question

using the knowledge of odd and even functions, find the Fourier series decomposition of

f(x)={2x,-π<x<π

{f(x+2π)

Answer 1

For the odd function, the Fourier series is called the Fourier Sine series and is given by

$$f_{odd}(x)=\displaystyle\sum_{n=1}^{\infin}b_n\sin nx$$

where the Fourier coefficients are

$$b_n=\dfrac{2}{\pi}\displaystyle\int_{0}^{\pi}f(x)\sin nxdx, n=1, 2, 3, ...$$

$$b_n=\dfrac{2}{\pi}\displaystyle\int_{0}^{\pi}2x\sin nxdx$$

$$\int 2x\sin nx dx$$

$$\int u dv=uv-\int v du$$

$$u=2x, du=2dx$$

$$dv=\sin nxdx, v=\int \sin nx dx=-\dfrac{1}{n}\cos nx$$

$$\int 2x\sin nx dx=-\dfrac{2x}{n}\cos nx+\int \dfrac{2}{n}\cos nxdx$$

$$=-\dfrac{2x}{n}\cos nx+\dfrac{2}{n^2}\sin nx+C$$

$$b_n=\dfrac{2}{\pi}\bigg[-\dfrac{2x}{n}\cos nx+\dfrac{2}{n^2}\sin nx\bigg]\begin{matrix} \pi \\ 0 \end{matrix}$$

$$=-\dfrac{4}{n}\cos \pi n=\dfrac{(-1)^{n+1}4}{n}$$

$$f(x)=\displaystyle\sum_{n=1}^{\infin}\dfrac{(-1)^{n+1}4}{n}\sin nx$$