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| # | Fourier series over original domain | Fourier series over periodic extended domain |
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| 1 |
\(f\left (x\right ) =\begin {cases} x & -2<x<0 \\ 1 & 0<x<2 \end {cases} \hspace {15pt} -2<x<2\)
\(f\left ( x\right ) \sim 2 \sum _{n=1}^{\infty } - \frac {2(-1+(-1)^n)}{n^2 \pi ^2} \cos (\frac {\pi }{2} n x) + \frac {1-3 (-1)^n}{n \pi } \sin (\frac {\pi }{2} n x)\) |
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| 2 |
\(f\left (x\right ) =-x \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim 2\sum _{n=1}^{\infty }\frac {\left ( -1\right ) ^{n}}{n}\sin \left ( nx\right )\) |
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| 3 |
\(f\left ( x\right ) =\left \{ \begin {array} [c]{ccc}x & &\hspace {8pt} \left \vert x\right \vert <\frac {\pi }{2}\\ 0 & &\hspace {8pt} \text { otherwise }\end {array} \right . \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim \sum _{n=1}^{\infty }\frac {2}{\pi n^{2}}\left ( \sin \left ( \frac {n\pi }{2}\right ) -\frac {1}{2}n\pi \cos \left ( \frac {n\pi }{2}\right ) \right ) \sin \left ( nx\right )\) |
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| 4 |
\(f\left ( x\right ) =\left \{ \begin {array} [c]{ccc} -1 & &\hspace {8pt} -\pi \leq x\leq 0\\ 1 & &\hspace {8pt} 0<x\leq \pi \end {array} \right . \hspace {15pt} -\pi <x<\pi \)
\(f(x) \sim \frac {4}{\pi } \sum _{n=1}^{\infty } \frac {1}{2 n- 1} \sin ( (2 n -1) x) \) |
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| 5 |
\(f\left ( x\right ) =\left \{ \begin {array} [c]{ccc} 1 & &\hspace {8pt} |x|< \frac {1}{2}\pi \\ 0 & &\hspace {8pt} \frac {1}{2}\pi <|x|< \pi \end {array} \right . \hspace {15pt} -\pi <x<\pi \)
\(f(x) \sim \frac {1}{2} + \sum _{n=1}^{\infty } \frac {2}{\pi n} \sin ( \frac {n \pi }{2} ) \cos (n x) \) |
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| 6 |
\(f\left ( x\right ) =x^{2} \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim \frac {1}{3}\pi ^{2}+4\sum _{n=1}^{\infty }\frac {1}{n^{2}}\left ( -1\right ) ^{n}\cos \left ( nx\right )\) |
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| 7 |
\(f\left ( x\right ) =\left \{ \begin {array} [c]{ccc}\sin x & & \hspace {8pt}0<x\leq \pi \\ 0 & & \hspace {8pt} -\pi \leq x<0 \end {array} \right . \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim \frac {1}{\pi }+\frac {1}{2}\sin \left ( x\right ) +\frac {2}{\pi }\sum _{n=1}^{\infty }\frac {1}{1-\left ( 2n\right ) ^{2}}\cos \left ( 2nx\right )\) |
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| 8 |
\(f\left ( x\right ) =\left \{ \begin {array} [c]{ccc}x^{2} & & \hspace {8pt} 0<x\leq \pi \\ -x^{2} & & \hspace {8pt} -\pi \leq x<0 \end {array} \right . \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim 2\pi ^{2}\sum _{n=1}^{\infty }\left ( \frac {1}{n\pi }\left ( -1\right ) ^{n+1}-\frac {2}{\left ( n\pi \right ) ^{3}}\left ( 1-\left ( -1\right ) ^{n}\right ) \right ) \sin \left ( nx\right )\) |
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| 9 |
\(f\left ( x\right ) =x+\frac {1}{4}x^{2} \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim \frac {\pi ^{2}}{12}+\sum _{n=1}^{\infty }\left ( -1\right ) ^{n}\left ( \frac {\cos \left ( nx\right ) }{n^{2}}-\frac {2\sin \left ( nx\right ) }{n}\right ) \) |
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| 10 |
\(f\left ( x\right ) =e^{x} \hspace {15pt} -\pi <x<\pi \)
\(f\left ( x\right ) \sim \frac {2\sinh \left ( \pi \right ) }{\pi }\left ( \frac {1}{2}+\sum _{n=1}^{\infty }\frac {\left ( -1\right ) ^{n}}{1+n^{2}}\left ( a\cos \left ( nx\right ) -n\sin \left ( nx\right ) \right ) \right ) \) |
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| 11 |
\(f(x) =\begin {cases} 1 & \left | x\right | <\frac {\pi }{2} \\ 0 & \text {True} \end {cases} \hspace {15pt} -\pi <x<\pi \)
\(f(x) \sim \frac {1}{2} + \sum _{n=1}^{\infty } \frac {2}{n \pi } \sin (\frac {n \pi }{2}) \cos (n x) \) |
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| 12 |
\(f(x) =x^3 \hspace {15pt} -\pi <x<\pi \)
\(f(x) \sim \sum _{n=1}^{\infty } - \frac { 2(-1)^n (-6 n^2 \pi ^2)}{n^3} \sin (n x) \) |
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| 13 |
\(f(x) =\sin (x) \hspace {15pt} -1<x<1\)
\(f(x) \sim \sum _{n=1}^{\infty } - \frac {2 n \pi (-1)^n }{n^2 \pi ^2 -1} \sin (1) \sin (n \pi x) \) |
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