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# Formelsammlung Mathematik: Unendliche Reihen: Fourierreihen

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##### 1
${\displaystyle \sum _{k=0}^{\infty }\varrho ^{k}\cos k\varphi ={\frac {1-\varrho \cos \varphi }{1-2\varrho \cos \varphi +\varrho ^{2}}}\qquad \sum _{k=0}^{\infty }\varrho ^{k}\sin k\varphi ={\frac {\varrho \sin \varphi }{1-2\varrho \cos \varphi +\varrho ^{2}}}\qquad |\varrho |<1\,,\,\varphi \in \mathbb {R} }$

##### 2
${\displaystyle \sum _{k=1}^{\infty }{\frac {\varrho ^{k}\cos k\varphi }{k}}=-{\frac {1}{2}}\,\log \left(1-2\varrho \cos \varphi \,+\varrho ^{2}\right)\qquad \sum _{k=1}^{\infty }{\frac {\varrho ^{k}\sin k\varphi }{k}}=\arctan \left({\frac {\varrho \,\sin \varphi }{1-\varrho \,\cos \varphi }}\right)\qquad -1<\varrho <1\;,\;\varphi \in \mathbb {R} }$

##### 3
${\displaystyle \sum _{k=1}^{\infty }{\frac {(-1)^{k}\cos k\varphi }{k}}=-\log \left(2\,\cos {\frac {\varphi }{2}}\right)\qquad \sum _{k=1}^{\infty }{\frac {(-1)^{k}\sin k\varphi }{k}}=-{\frac {\varphi }{2}}\qquad -\pi <\varphi <\pi }$

##### 4
${\displaystyle \sum _{k=1}^{\infty }{\frac {\cos k\varphi }{k}}=-\log \left(2\,\sin {\frac {\varphi }{2}}\right)\qquad \sum _{k=1}^{\infty }{\frac {\sin k\varphi }{k}}={\frac {\pi -\varphi }{2}}\qquad 0<\varphi <2\pi }$

##### 5
${\displaystyle \sum _{k=0}^{\infty }{\frac {\cos(2k+1)\varphi }{2k+1}}=-{\frac {1}{2}}\log \left(\tan {\frac {\varphi }{2}}\right)\qquad \sum _{k=0}^{\infty }{\frac {\sin(2k+1)\varphi }{2k+1}}={\frac {\pi }{4}}\qquad 0<\varphi <\pi }$

##### 6
${\displaystyle \sum _{k=0}^{\infty }{\frac {(-1)^{k}\,\cos(2k+1)\varphi }{2k+1}}={\frac {\pi }{4}}\qquad \sum _{k=0}^{\infty }{\frac {(-1)^{k}\,\sin(2k+1)\varphi }{2k+1}}={\frac {1}{2}}\log \left(\tan \left({\frac {\pi }{4}}+{\frac {\varphi }{2}}\right)\right)\qquad -{\frac {\pi }{2}}<\varphi <{\frac {\pi }{2}}}$

##### 7
${\displaystyle \sum _{k\in \mathbb {Z} }{\frac {(-1)^{k}\,\cos kx}{k+\alpha }}=\pi \,{\frac {\cos \alpha x}{\sin \alpha \pi }}\qquad \sum _{k\in \mathbb {Z} }{\frac {(-1)^{k}\,\sin kx}{k+\alpha }}=-\pi \,{\frac {\sin \alpha x}{\sin \alpha \pi }}\qquad -\pi

##### 8
${\displaystyle \sum _{k\in \mathbb {Z} }{\frac {(-1)^{k}\,\cos(kx)}{k^{2}+\alpha ^{2}}}={\frac {\pi }{\alpha }}\,{\frac {\cosh(\alpha x)}{\sinh(\alpha \pi )}}\qquad -\pi

##### 9
${\displaystyle {\frac {2}{\pi }}+{\frac {2}{\pi }}\sum _{n=1}^{\infty }\left({\frac {1}{2n+1}}-{\frac {1}{2n-1}}\right)\,\cos 2nx=|\sin x|\qquad x\in \mathbb {R} }$

##### 10
${\displaystyle {\frac {2}{\pi }}+{\frac {2}{\pi }}\sum _{n=1}^{\infty }(-1)^{n}\left({\frac {1}{2n+1}}-{\frac {1}{2n-1}}\right)\,\cos 2nx=|\cos x|\qquad x\in \mathbb {R} }$

##### 11
Besitzt die Funktion ${\displaystyle f\,}$ die reelle Fourierreihenentwicklung ${\displaystyle f(x)={\frac {a_{0}}{2}}+\sum _{k=1}^{\infty }\left(a_{k}\cos kx+b_{k}\sin kx\right)}$, so gilt ${\displaystyle {\frac {1}{\pi }}\int _{-\pi }^{\pi }|f(x)|^{2}\,dx={\frac {a_{0}^{2}}{2}}+\sum _{k=1}^{\infty }\left(a_{k}^{2}+b_{k}^{2}\right)}$.