# Fourier Series

The formula for a Fourier series on an interval [c,c+T] is:

$\displaystyle f(x) = {a_0\over 2} + \sum_{n=1}^\infty \left[a_n\cos\left({2n\pi x \over T}\right) + b_n\sin\left({2n\pi x\over T}\right)\right]\,$

$\displaystyle a_n = {2\over T}\int_c^{c+T} f(x) \cos\left({2n\pi x\over T}\right)\,dx\,$

$\displaystyle b_n = {2\over T}\int_c^{c+T} f(x) \sin\left({2n\pi x\over T}\right)\,dx\,$

1. solution Find the Fourier series for $\displaystyle |x|\,$ , $\displaystyle - \pi < x < \pi \,$

2. solution Find the Fourier series for $\displaystyle f(x) = \begin{cases}0 & -\pi < x < 0,\\ 1 & 0 < x < \pi\end{cases}$

3. solution Find the Fourier series for $\displaystyle 1+x\,$ on $\displaystyle [-\pi,\pi]\,$

4. solution Find the Fourier series for $\displaystyle f(x) = \begin{cases}1 & -1 \le x < 0\\ \frac{1}{2} & x = 0\\x&0 on $\displaystyle [-1,1]\,$

5. solution Find the Fourier series for $\displaystyle f(x) = \begin{cases}-1 & -3 \le x < 0\\ 1&0 on $\displaystyle [-3,3]\,$

6. solution Find the Fourier series for $\displaystyle x^2\,$ on $\displaystyle [-\pi,\pi]\,$

7. solution Find the Fourier series for a function $\displaystyle f(x) = f(x+2), f(x) = (x-1)(x-3)\,$ on $\displaystyle [1,3]\,$ .

8. solution Find the Fourier series for $\displaystyle f(x) = x\,$ on $\displaystyle [0,1]\,$ .

9. solution Find the Fourier series for $\displaystyle f(t) = \begin{cases}\frac{4}{\pi}t & 0 \le t < \frac{\pi}{2},\\ \frac{-4}{\pi}t & \frac{-\pi}{2} \le t \le 0\end{cases}$