Formelsammlung Mathematik: Unendliche Reihen: Fourierreihen

Zurück zu Unendliche Reihen

1

$\sum _{k=0}^{\mathrm{\infty }}{\varrho }^k\cos k\phi =\frac{1-\varrho \cos \phi }{1-2\varrho \cos \phi +{\varrho }^2}\sum _{k=0}^{\mathrm{\infty }}{\varrho }^k\sin k\phi =\frac{\varrho \sin \phi }{1-2\varrho \cos \phi +{\varrho }^2}|\varrho |<1,\phi \in ℝ$

2

$\sum _{k=1}^{\mathrm{\infty }}\frac{{\varrho }^k\cos k\phi }{k}=-\frac{1}{2}\log (1-2\varrho \cos \phi +{\varrho }^2)\sum _{k=1}^{\mathrm{\infty }}\frac{{\varrho }^k\sin k\phi }{k}=\arctan \left(\frac{\varrho \sin \phi }{1-\varrho \cos \phi }\right)-1<\varrho <1,\phi \in ℝ$

3

$\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k\cos k\phi }{k}=-\log (2\cos \frac{\phi }{2})\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^k\sin k\phi }{k}=-\frac{\phi }{2}-\pi <\phi <\pi$

4

$\sum _{k=1}^{\mathrm{\infty }}\frac{\cos k\phi }{k}=-\log (2\sin \frac{\phi }{2})\sum _{k=1}^{\mathrm{\infty }}\frac{\sin k\phi }{k}=\frac{\pi -\phi }{2}0<\phi <2\pi$

5

$\sum _{k=0}^{\mathrm{\infty }}\frac{\cos (2k+1)\phi }{2k+1}=-\frac{1}{2}\log (\tan \frac{\phi }{2})\sum _{k=0}^{\mathrm{\infty }}\frac{\sin (2k+1)\phi }{2k+1}=\frac{\pi }{4}0<\phi <\pi$

6

$\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\cos (2k+1)\phi }{2k+1}=\frac{\pi }{4}\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k\sin (2k+1)\phi }{2k+1}=\frac{1}{2}\log (\tan (\frac{\pi }{4}+\frac{\phi }{2}))-\frac{\pi }{2}<\phi <\frac{\pi }{2}$

7

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

8

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

9

$\frac{2}{\pi }+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}(\frac{1}{2n+1}-\frac{1}{2n-1})\cos 2nx=|\sin x|x\in ℝ$

10

$\frac{2}{\pi }+\frac{2}{\pi }\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n(\frac{1}{2n+1}-\frac{1}{2n-1})\cos 2nx=|\cos x|x\in ℝ$

11

Besitzt die Funktion $f$ die reelle Fourierreihenentwicklung $f(x)=\frac{a_0}{2}+\sum _{k=1}^{\mathrm{\infty }}(a_k\cos kx+b_k\sin kx)$, so gilt $\frac{1}{\pi }\underset{-\pi }{\overset{\pi }{\int }}|f(x){|}^{2}dx=\frac{a_0^2}{2}+\sum _{k=1}^{\mathrm{\infty }}(a_k^2+b_k^2)$.