Answer to Question #218954 in Calculus for hellena

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

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

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

$\int 2x\sin nx dx$

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

$u=2x, du=2dx$

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

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

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

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

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

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