Expand the following function in a Fourier series...
f(x)=x (0 < x < 2pi)
These fourier series make no freaking sense to me. Please help me.
Is...
$\displaystyle f(x) = \frac {a_{0}}{2} + \sum_{n=1} ^{\infty} a_{n} \cdot \cos n x + b_{n}\cdot \sin nx$ (1)
... where...
$\displaystyle a_{n} = \frac{1}{\pi} \int _{0}^{2\pi} f(x)\cdot \cos nx\cdot dx$
$\displaystyle b_{n} = \frac{1}{\pi} \int _{0}^{2\pi} f(x)\cdot \sin nx\cdot dx$ (2)
In your case is $\displaystyle f(x)=x$ so that...
$\displaystyle a_{0} = 2 \pi$
$\displaystyle a_{n} =0, n\ge 1$
$\displaystyle b_{n} = - \frac {2}{n}, n \ge 1$ (3)
Therefore the (1) becomes...
$\displaystyle f(x) = \pi - 2\cdot \sum _{n=1}^{\infty} \frac {\sin nx}{n}$ (4)
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$