Fourier Series

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

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]\,

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

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 |x|\,, - \pi < x < \pi \,

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

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

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

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

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

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


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

9. solution Find the Fourier series for 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}



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