# 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

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

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\leq 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\leq 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 ${\displaystyle f(t)={\begin{cases}{\frac {4}{\pi }}t&0\leq t<{\frac {\pi }{2}},\\{\frac {-4}{\pi }}t&{\frac {-\pi }{2}}\leq t\leq 0\end{cases}}}$