1. ## find Fourier coefficients

Fourier coefficients

2. Let's see. It's been a while since I worked a Fourier.

$\displaystyle \begin{Bmatrix}f(x)=\frac{x}{2}, \;\ {-\pi}<x<0\\f(x)=0, \;\ 0<x<{\pi}\end{Bmatrix}$

$\displaystyle a_{0}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}dx=\frac{-\pi}{4}$

$\displaystyle a_{n}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}cos(nx)dx$

=$\displaystyle \frac{-cos(n\pi)}{2n^{2}\pi}-\frac{sin(n\pi)}{2n}+\frac{1}{2n^{2}\pi}$

But $\displaystyle cos(n\pi)=(-1)^{n} \;\ and \;\ sin(n\pi)=0$

So, we have $\displaystyle a_{n}=\frac{1-(-1)^{n}}{2n^{2}\pi}$

$\displaystyle b_{n}=\frac{1}{\pi}\int_{-\pi}^{0}\frac{x}{2}sin(nx)dx$

=$\displaystyle \frac{sin(n\pi)}{2n^{2}\pi}-\frac{cos(n\pi)}{2n}=\frac{-(-1)^{n}}{2n}$

Check and make sure I didn't err somewhere.