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de:Fourierreihe (Absatz) innen van.
h
{\displaystyle h}
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sin
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3
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sin
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{\displaystyle \displaystyle f(t)={\frac {8h}{\pi ^{2}}}\left[{\sin {\omega t}-{\frac {1}{3^{2}}}\sin {3\omega t}+{\frac {1}{5^{2}}}\sin {5\omega t}-\ldots }\right]={\frac {4h}{\pi ^{2}}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {\sin((2k-1)\omega t)}{(2k-1)^{2}}}}
Gleiches gilt für den Rechteckimpuls:
f
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4
h
π
[
sin
ω
t
+
1
3
sin
3
ω
t
+
1
5
sin
5
ω
t
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]
=
4
h
π
∑
k
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1
∞
sin
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2
k
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ω
t
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2
k
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1
{\displaystyle \displaystyle f(t)={\frac {4h}{\pi }}\left[{\sin {\omega t}+{\frac {1}{3}}\sin {3\omega t}+{\frac {1}{5}}\sin {5\omega t}+\ldots }\right]={\frac {4h}{\pi }}\sum _{k=1}^{\infty }{\sin {\left((2k-1)\omega t\right)} \over 2k-1}}
Ebenso lassen sich punktsymmetrische Funktionen aus Sinustermen approximieren. Hier erreicht man eine Phasenverschiebung durch alternierende Vorzeichen:
f
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h
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[
sin
ω
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1
2
sin
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sin
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ω
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k
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sin
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ω
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k
{\displaystyle \displaystyle f(t)=-{\frac {2h}{\pi }}\left[{\sin {\omega t}-{\frac {1}{2}}\sin {2\omega t}+{\frac {1}{3}}\sin {3\omega t}-\ldots }\right]=-{\frac {2h}{\pi }}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {\sin k\omega t}{k}}}
f
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h
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sin
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1
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cos
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t
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cos
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ω
t
15
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cos
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35
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.
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.
]
=
2
h
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4
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cos
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{\displaystyle \displaystyle f(t)=h\left|\sin {\omega t}\right|={\frac {4h}{\pi }}\left[{\frac {1}{2}}-{\frac {\cos {2\omega t}}{3}}-{\frac {\cos {4\omega t}}{15}}-{\frac {\cos {6\omega t}}{35}}-...\right]={\frac {2h}{\pi }}-{\frac {4h}{\pi }}\sum _{k=1}^{\infty }{\frac {\cos {2k\omega t}}{(2k)^{2}-1}}}
![{\displaystyle \displaystyle f(t)={\frac {8h}{\pi ^{2}}}\left[{\sin {\omega t}-{\frac {1}{3^{2}}}\sin {3\omega t}+{\frac {1}{5^{2}}}\sin {5\omega t}-\ldots }\right]={\frac {4h}{\pi ^{2}}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {\sin((2k-1)\omega t)}{(2k-1)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8b42268aad64aa2308f03317606ed4e111102d)
![{\displaystyle \displaystyle f(t)={\frac {4h}{\pi }}\left[{\sin {\omega t}+{\frac {1}{3}}\sin {3\omega t}+{\frac {1}{5}}\sin {5\omega t}+\ldots }\right]={\frac {4h}{\pi }}\sum _{k=1}^{\infty }{\sin {\left((2k-1)\omega t\right)} \over 2k-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f306c7eac89557d3866df3f4325b928b2a6c11)
![{\displaystyle \displaystyle f(t)=-{\frac {2h}{\pi }}\left[{\sin {\omega t}-{\frac {1}{2}}\sin {2\omega t}+{\frac {1}{3}}\sin {3\omega t}-\ldots }\right]=-{\frac {2h}{\pi }}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {\sin k\omega t}{k}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44dd83a47c6cfed434451dd66d9be2d26f0872b2)
![{\displaystyle \displaystyle f(t)=h\left|\sin {\omega t}\right|={\frac {4h}{\pi }}\left[{\frac {1}{2}}-{\frac {\cos {2\omega t}}{3}}-{\frac {\cos {4\omega t}}{15}}-{\frac {\cos {6\omega t}}{35}}-...\right]={\frac {2h}{\pi }}-{\frac {4h}{\pi }}\sum _{k=1}^{\infty }{\frac {\cos {2k\omega t}}{(2k)^{2}-1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a5390da43d47b6f564a01b4d1e8e25d2027d519)