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\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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