Fourier Series

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

$f(x)={a_{0} \over 2}+\sum _{{n=1}}^{\infty }\left[a_{n}\cos \left({2n\pi x \over T}\right)+b_{n}\sin \left({2n\pi x \over T}\right)\right]\,$

$a_{n}={2 \over T}\int _{c}^{{c+T}}f(x)\cos \left({2n\pi x \over T}\right)\,dx\,$

$b_{n}={2 \over T}\int _{c}^{{c+T}}f(x)\sin \left({2n\pi x \over T}\right)\,dx\,$

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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 $f(x)={\begin{cases}1&-1\leq x<0\\{\frac {1}{2}}&x=0\\x&0<x\leq 1\end{cases}}$ on $[-1,1]\,$

5. solution Find the Fourier series for $f(x)={\begin{cases}-1&-3\leq x<0\\1&0<x\leq 3\end{cases}}$ 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 $f(t)={\begin{cases}{\frac {4}{\pi }}t&0\leq t<{\frac {\pi }{2}},\\{\frac {-4}{\pi }}t&{\frac {-\pi }{2}}\leq t\leq 0\end{cases}}$

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