Formelsammlung Mathematik: Unendliche Reihen

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Inhaltsverzeichnis

[Bearbeiten] Unendliche geometrische Reihe


\sum_{k=0}^\infty z^k=\frac{1}{1-z} \qquad |z|<1\!



[Bearbeiten] Binomische Reihe


(1+z)^\alpha=\sum_{k=0}^\infty {\alpha \choose k} z^k \qquad |z|<1 \; , \; \alpha\in\Bbb{C}



[Bearbeiten] Reihe mit Lambert W-Funktion


\sum_{k=0}^\infty \frac{k^k}{k!}\, z^k=\frac{1}{1+W(-z)} \qquad |z|\le\frac{1}{e} \; , \; z\neq \frac{1}{e}



[Bearbeiten] Catalansche Konstante


G=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}


[Bearbeiten] Erste Vacca'sche Reihe


\sum_{k=2}^\infty \frac{(-1)^k\,\lfloor \log_2 k \rfloor}{k}=\gamma



[Bearbeiten] Zweite Vacca'sche Reihe


\sum_{k=1}^\infty \left(\frac{1}{\lfloor \sqrt{k} \rfloor^2}-\frac{1}{k}\right)=\gamma+\frac{\pi^2}{6}



[Bearbeiten] Reihen mit zentrierten Binomialkoeffizienten


\sum_{k=1}^\infty \frac{1}{{2k\choose k}}=\frac{2\pi\sqrt{3}+9}{27}


\sum_{k=1}^\infty \frac{1}{k\, {2k\choose k}}=\frac{\pi}{3\sqrt{3}}


\sum_{k=1}^\infty \frac{1}{k^2\, {2k\choose k}}=\frac13\zeta(2)


\sum_{k=1}^\infty \frac{1}{k^4\, {2k\choose k}}=\frac{17}{36}\zeta(4)



\sum_{k=1}^\infty \frac{(-1)^{k-1}}{{2k\choose k}}=\frac{4\sqrt{5}\log\phi+5}{25}


\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k\, {2k\choose k}}=\frac{2}{\sqrt{5}}\log\phi


\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^2\, {2k\choose k}}=2\log^2\phi



\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k^3\, {2k\choose k}}=\frac25 \zeta(3)




[Bearbeiten] Reihen zur Jacobischen Thetafunktion


Es sei \vartheta_3(q)=\sum_{n\in\Bbb{Z}} q^{n^2} und a_m:=\frac{\Gamma\left(\frac34\right)}{\pi^\frac14}\,\vartheta_3\left(e^{-m\pi}\right). Vermutlich wird a_m\, algebraisch sein \forall m\in\Bbb{Z}^{>0}.
m\,
Minimalpolynom m\, von a_m\, \text{deg}\, m\, a_m\, ausgedrückt durch Radikale
1\, x-1\, 1\, 1\,
2\, 8x^4-8x^2+1\, 4\, \frac{\sqrt{\sqrt{2}+2}}{2}
3\, 27x^8-18x^4-1\, 8\, \sqrt[4]{\frac{2+\sqrt{3}}{3\sqrt{3}}}
4\, 32x^4-64x^3+48x^2-16x+1\, 4\, \frac{2+\sqrt[4]{2}^3}{4}
5\, 25x^4-20x^2-1\, 4\, \sqrt{\frac{\sqrt{5}+2}{5}}
6\, \frac{\sqrt[3]{-4+3\sqrt{2}+\sqrt[4]{3}^5+2\sqrt{3}-\sqrt[4]{3}^3+2\sqrt{2}\sqrt[4]{3}^3}}{2\, \sqrt[8]{3}^3\, \sqrt[6]{(\sqrt{2}-1)(\sqrt{3}-1)}}
7\, 823543 x^{16}-470596 x^{12}-72030 x^8-10388 x^4-1\, 16\, \frac{\sqrt[4]{7+4\sqrt{7}+2\sqrt{35+16\sqrt{7}}}}{\sqrt{7}}
8\, 268435456x^{16}-268435456x^{14}+...-84608x^2+1\, 16\,
9\, 243x^6-486x^5+405x^4-216x^3+81x^2-18x-1\, 6\, \frac{1+\sqrt[3]{2+2\sqrt{3}}}{3}
10\, 1600000000x^{16}-2560000000x^{14}+...-1958080x^2+1\, 16\,
12\, 32\,
\vartheta_3\left(e^{-\sqrt{2}\pi}\right)=\frac{\Gamma\left(\frac98\right)}{\Gamma\left(\frac54\right)}\, \sqrt{\frac{\Gamma\left(\frac14\right)}{\sqrt[4]{2}\,\pi}}


[Bearbeiten] Hypergeometrische Reihen


Formel von Gauß
\sum_{k=0}^\infty \frac{\Gamma(a+k)\, \Gamma(b+k)}{k!\, \Gamma(c+k)}=\frac{\Gamma(a)\,\Gamma(b)\, \Gamma(c-a-b)}{\Gamma(c-a)\, \Gamma(c-b)} \qquad \text{Re}(c-a-b)>0



Formel von Gauß (modifiziert)
\sum_{k=0}^\infty \frac{1}{k!\, (\alpha+k)!\, (\beta-k)!\, (\gamma-k)!}=\frac{(\alpha+\beta+\gamma)!}{\beta!\,\gamma!\, (\alpha+\beta)!\, (\alpha+\gamma)!} \qquad \text{Re}(\alpha+\beta+\gamma)>-1



Kummer'sche Formel
\sum_{n=0}^\infty (-1)^n\, \frac{\Gamma(a+n)\,\Gamma(b+n)}{n!\; \Gamma(1+a-b+n)}=\frac{\Gamma\left(\frac{a}{2}\right)\,\Gamma(b)}{2\,\Gamma\left(1+\frac{a}{2}-b\right)}



Dougall'sche Formel
\sum_{n=-\infty}^\infty \frac{\Gamma(a+n)\, \Gamma(b+n)}{\Gamma(c+n)\, \Gamma(d+n)}=\frac{\pi^2}{\sin a\pi\, \sin b\pi}\,\frac{\Gamma(c+d-a-b-1)}{\Gamma(c-a)\, \Gamma(d-a)\, \Gamma(c-b)\, \Gamma(d-b)} \qquad \begin{matrix}\text{Re}(c+d-a-b)>1 \; , \; a,b\in\Bbb{C}\setminus\Bbb{Z} \\\\ c-a\; ,\; d-a\; ,\; c-b \; ,\; d-b\notin\Bbb{Z}^{\le 0}\end{matrix}



Formel von Dixon
\sum_{n=0}^\infty \frac{\Gamma(a+n)\,\Gamma(b+n)\,\Gamma(c+n)}{n!\; \Gamma(1+a-b+n)\, \Gamma(1+a-c+n)}=\frac{\Gamma\left(\frac{a}{2}\right) \Gamma(b)\,\Gamma(c)\,\Gamma\left(1+\frac{a}{2}-b-c\right)}{2\;\Gamma\left(1+\frac{a}{2}-b\right)\,\Gamma\left(1+\frac{a}{2}-c\right)\,\Gamma\left(1+a-b-c\right)}



[Bearbeiten] Reihen zur Betafunktion


\sum_{k=0}^\infty B(x+k,y+1)=B(x,y)



\sum_{k=0}^\infty (-1)^k\, {x-1\choose k}\, \frac{1}{k+y}=B(x,y) \qquad \text{Re}(x)>0\, , \, y\notin\Bbb{Z}^{\le 0}



[Bearbeiten] Reihen zur Psifunktion


\sum_{k=0}^\infty \frac{(-1)^k}{\alpha+\beta\, k}=\frac{1}{2\beta}\, \left[\psi\left(\frac12+\frac{\alpha}{2\beta}\right)-\psi\left(\frac{\alpha}{2\beta}\right)\right]


\sum_{k=0}^\infty \frac{(-1)^k}{3k+1}=\frac{\log 2}{3}+\frac{\pi}{3\sqrt{3}}


\sum_{k=0}^\infty \frac{(-1)^k}{4k+1}=\frac{\log(1+\sqrt{2})}{2\sqrt{2}}+\frac{\pi}{4\sqrt{2}}


\sum_{k=0}^\infty \frac{(-1)^k}{5k+1}=\frac{\pi}{10}\csc\left(\frac{\pi}{5}\right)+\frac{1}{\sqrt{5}}\log\phi+\frac{\log 2}{5}


[Bearbeiten] Reihen deren Glieder ein Produkt von benachbarten Faktoren enthalten


\sum_{k=n}^\infty \frac{1}{k\, (k+1)}=\frac{1}{n}


\sum_{k=1}^\infty \frac{(1-z)^{k+n}\, (-1)^n}{k\,(k+1)\cdots (k+n)}
=\frac{z^n}{n!}\, (H_n-\log z)+
\sum_{k=1}^n \frac{(-1)^k}{k!\, k}\,\frac{z^{n-k}}{(n-k)!}


\sum_{k=1}^\infty \frac{1}{k\, (k+1)\cdots (k+n)}=\frac{1}{n\cdot n!}




\sum_{k=1}^\infty \frac{1}{k\,(k+2)\, (k+4)\cdots (k+2n)}=\frac{1}{2n}\left(\frac{2^n\, n!}{(2n)!}+\frac{1}{2^n\, n!}\right)


\sum_{k=1}^\infty \frac{1}{k^2\, (k+1)^2 \cdots (k+n)^2}=\frac{1}{n!^2} {2n\choose n} \left(\frac{\pi^2}{6}-3\sum_{k=1}^n \frac{1}{k^2 {2k\choose k}}\right)


\sum_{k=0}^\infty \frac{1}{(2k+1)^2\, (2k+3)^2\cdots (2k+2n+1)^2}=\frac{1}{n!^2} \frac{{2n\choose n}}{2^{2n}} \left(
\frac{\pi^2}{8}-\frac14 \sum_{k=1}^n \frac{(3k-1) \,2^{4k}}{k^3 {2k\choose k}^3} \right)


\sum_{k=1}^\infty \frac{1}{k^3\, (k+1)^3\cdots (k+2n)^3}=\frac{(-1)^n\, (3n)!}{(2n)!^3\, n!^3}\left(\zeta(3)+\frac14\sum_{k=1}^n (-1)^k\, \frac{(56k^2-32k+5)\, (k-1)!^3}{(3k)!\, (2k-1)^2}\right)


[Bearbeiten] Reihe die sich aus der geometrischen Reihe ergibt


\sum_{k=0}^\infty k^n \, x^k=\sum_{k=0}^n \begin{Bmatrix} n \\ k\end{Bmatrix} \frac{k! \, x^k}{(1-x)^{k+1}} \qquad |x|<1



[Bearbeiten] Reihen zur Zetafunktion


\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}



\zeta(2n)=\sum_{k=1}^\infty \frac{1}{k^{2n}}=\frac{(-1)^{n-1}\, B_{2n}\, \pi^{2n}\, 2^{2n-1}}{(2n)!}



[Bearbeiten] Reihen zur Dirichlet Betafunktion


\beta(2n+1)=\sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^{2n+1}}=|E_{2n}|\,\frac{\pi^{2n+1}}{2^{2n+2}\, (2n)!}




\beta'(1)=\sum_{k=1}^\infty \frac{(-1)^{k+1} \, \log(2k+1)}{2k+1}=\frac{\pi \gamma}{4}+\frac{\pi}{2} \log\left(\sqrt{2\pi}\,\,\frac{\Gamma\left(\frac34\right)}{\Gamma\left(\frac14\right)}\right)




[Bearbeiten] Reihen zur Dirichlet Etafunktion


\eta'(1)=\sum_{k=1}^\infty \frac{(-1)^k \, \log k}{k}=\gamma\, \log 2-\frac12 \log^2 2



(-1)^{n-1}\, \eta^{(n)}(1)=\sum_{k=1}^\infty \frac{(-1)^k \, (\log k)^n}{k}=-\frac{(\log 2)^{n+1}}{n+1}+\sum_{k=0}^{n-1} {n\choose k}\, \gamma_k\, (\log 2)^{n-k}



[Bearbeiten] Fourier-Reihen für Bernoulli- und Euler-Polynome


\sum_{k=1}^\infty \frac{\cos\left(2k\pi x-\frac{n\pi}{2}\right)}{k^n}=-\frac{B_n(x)\, (2\pi)^n}{2\, n!}


\sum_{k=0}^\infty \frac{\cos\left((2k+1)\pi x-\frac{n\pi}{2}\right)}{(2k+1)^n}=\frac{E_{n-1}(x)\, \pi^n}{4\, (n-1)!}


[Bearbeiten] Reihen die sich mit Residuensatz und Kettenbruchentwicklung berechnen lassen


Ist m\in\Bbb{Z}^{\ge 2} und ist \alpha_0\, ein Element des quadratischen Zahlkörpers \Bbb{Q}(\sqrt{d}), so ist auch \sum_{n=1}^\infty \frac{\cot(n\pi\alpha_0)}{(n\pi)^{2m-1}}\in\Bbb{Q}(\sqrt{d}).



\sum_{k=1}^\infty \frac{\csc(n\pi \sqrt{2})}{(n\pi)^3}=-\frac{13}{360\sqrt{2}}


\sum_{n=1}^\infty \frac{\cot(n\pi \phi)}{(n\pi)^3}=-\frac{1}{45\sqrt{5}}



[Bearbeiten] Reihen die sich aus Partialbruchentwicklungen ergeben


\sum_{k\in\Bbb{Z}} \frac{\frac{\alpha^2}{2}}{k^4+\left(\frac{\alpha^2}{2}\right)^2}=\frac{\pi}{\alpha}\, \frac{\sinh\alpha\pi+\sin\alpha\pi}{\cosh\alpha\pi-\cos\alpha \pi}


\sum_{k\in\Bbb{Z}} \frac{k^2}{k^4+\left(\frac{\alpha^2}{2}\right)^2}=\frac{\pi}{\alpha}\, \frac{\sinh\alpha\pi-\sin\alpha\pi}{\cosh\alpha\pi-\cos\alpha \pi}


[Bearbeiten] Erzeugende Funktion der harmonischen Zahlen


\sum_{n=1}^\infty H_n \, x^n=-\frac{\log(1-x)}{1-x} \qquad |x|<1



[Bearbeiten] Reihen mit harmonischen Zahlen


\sum_{k=1}^\infty \frac{H_k}{k^2}=2\,\zeta(3)


\sum_{k=1}^\infty \frac{H_k}{k^3}=\frac{\pi^4}{72}


2\,\sum_{k=1}^\infty \frac{H_k}{k^n}=(n+2)\, \zeta(n+1)-\sum_{k=1}^{n-2} \zeta(n-k)\, \zeta(k+1) \qquad n\in\Bbb{Z}^{\ge 2}



\sum_{k=1}^\infty \frac{H_k}{k\, 2^k}=\frac{\pi^2}{12}


\sum_{k=1}^\infty \frac{H_k}{k^2\, 2^k}=\zeta(3)-\frac{\pi^2}{12}\log 2


\sum_{k=1}^\infty \frac{H_k^2}{k^2}=\frac{17\pi^4}{360}


\sum_{k=1}^\infty \frac{H_k^2}{(k+1)^2}=\frac{11\pi^4}{360}


[Bearbeiten] Knuthsche Reihe


\sum_{k=1}^\infty \left(\frac{k^k}{e^k\, k!}-\frac{1}{\sqrt{2\pi k}}\right)=-\frac23-\frac{1}{\sqrt{2\pi}}\, \zeta\left(\frac12\right)


[Bearbeiten] Ramanujan-Reihen


\sum_{k=1}^\infty \frac{\coth(k\pi)}{(k\pi)^{4n-1}}=\sum_{k=0}^{2n} (-1)^{k-1}\, \frac{\zeta(2k)}{\pi^{2k}}\, \frac{\zeta(4n-2k)}{\pi^{4n-2k}} \qquad n\in\Bbb{Z}^{>0}




\sum_{k=1}^\infty \frac{(-1)^k\,\text{csch}(k\pi)}{(k\pi)^{4n-1}}=\sum_{k=0}^{2n} (-1)^{k-1}\, \frac{\eta(2k)}{\pi^{2k}}\, \frac{\eta(4n-2k)}{\pi^{4n-2k}} \qquad n\in\Bbb{Z}


\sum_{k=0}^\infty \frac{\tanh\left((2k+1)\frac{\pi}{2}\right)}{\left((2k+1)\frac{\pi}{2}\right)^{4n-1}}=\sum_{k=0}^{2n} (-1)^{k-1}\, \frac{\lambda(2k)}{\left(\frac{\pi}{2}\right)^{2k}}\, \frac{\lambda(4n-2k)}{\left(\frac{\pi}{2}\right)^{4n-2k}} \qquad n\in\Bbb{Z}^{>0}


\sum_{k=0}^\infty \frac{(-1)^k\, \text{sech}\left((2k+1)\frac{\pi}{2}\right)}{\left((2k+1)\frac{\pi}{2}\right)^{4n+1}}=\sum_{k=0}^{2n} (-1)^k\, \frac{\beta(2k+1)}{\left(\frac{\pi}{2}\right)^{2k+1}}\, \frac{\beta(4n-2k+1)}{\left(\frac{\pi}{2}\right)^{4n-2k+1}} \qquad n\in\Bbb{Z}


\sum_{k=0}^\infty \frac{(-1)^k\, \text{sech}\left(\sqrt{3}\, (2k+1)\frac{\pi}{2}\right)}{(2k+1)\frac{\pi}{2}}=\frac{1}{12}


\sum_{k=0}^\infty \frac{(-1)^k\, \text{sech}\left(\frac{1}{\sqrt{3}}\, (2k+1)\frac{\pi}{2}\right)}{(2k+1)\frac{\pi}{2}}=\frac{5}{12}


\sum_{k=0}^\infty \frac{(-1)^k\, \text{sech}\left(\sqrt{3}\, (2k+1)\frac{\pi}{2}\right)}{\left((2k+1)\frac{\pi}{2}\right)^7}=\frac{1}{180}


\sum_{k=0}^\infty \frac{(-1)^k\, \text{sech}\left(\frac{1}{\sqrt{3}}\, (2k+1)\frac{\pi}{2}\right)}{\left((2k+1)\frac{\pi}{2}\right)^7}=\frac{143}{27\cdot 180}


\sum_{k=0}^\infty \frac{(2k+1)^{4n+1}}{e^{(2k+1)\pi}+1}=\left(2^{4n+1}-1\right)\, \frac{B_{4n+2}}{4\, (2n+1)} \qquad n\in\Bbb{Z}^{\ge 0}


\sum_{k=1}^\infty \frac{k^{-4n+1}}{e^{2k\pi}-1}=-\frac{(2\pi)^{4n-1}}{4}\, \sum_{k=0}^{2n} (-1)^k\, \frac{B_{2k}}{(2k)!}\, \frac{B_{4n-2k}}{(4n-2k)!}-\frac12 \, \zeta(4n-1) \qquad n\in\Bbb{Z}^{\ge 0}


\sum_{k=1}^\infty \frac{k^{4n+1}}{e^{2k\pi}-1}=\frac{B_{4n+2}}{4\, (2n+1)} \qquad n\in\Bbb{Z}^{>0}


\sum_{k=1}^\infty \frac{1}{k\, (e^{2\pi k}-1)}=\frac14 \log\left(\frac{4}{\pi}\right)-\frac{\pi}{12}+\log\Gamma\left(\frac34\right)


\sum_{k=1}^\infty \frac{k}{(-1)^k\, e^{\sqrt{3}\pi k}-1}=\frac{1}{24}-\frac{1}{4\sqrt{3}\pi}


\sum_{k\in\Bbb{Z}} \frac{1}{\cosh(k\pi)}=\frac{1}{\sqrt{2\pi}}\, \frac{\Gamma\left(\frac14\right)}{\Gamma\left(\frac34\right)}


\sum_{k\in\Bbb{Z}} \frac{\pi}{\cosh^2\left((2k+1)\frac{\pi}{2}\right)}=1


\sum_{k=1}^\infty \frac{(-1)^k}{k\, \sinh(k\pi)}=\frac{\pi}{12}-\frac12 \log 2


\sum_{k=1}^\infty \frac{1}{\sinh^2(k\pi)}=\frac16-\frac{1}{2\pi}


\sum_{k=1}^\infty \frac{1}{\sinh(2^k\pi)}=\coth\pi-1


\sum_{k=0}^\infty {-\frac12 \choose k}^3=\left(\frac{\Gamma\left(\frac98\right)}{\Gamma\left(\frac54\right)\, \Gamma\left(\frac78\right)}\right)^2


\sum_{k=0}^\infty (4k+1) {-\frac12 \choose k}^3=\frac{2}{\pi}


\sum_{k=0}^\infty (8k+1) {-\frac14 \choose k}^4=\frac{2\sqrt{2}}{\Gamma\left(\frac12\right)\,\Gamma^2\left(\frac34\right)}


\sum_{k=0}^\infty (4k+1) {-\frac12 \choose k}^5=\frac{2}{\Gamma^4\left(\frac34\right)}


[Bearbeiten] Polylogarithmus-Reihen


\text{Li}_2\left(\frac12\right)=\sum_{k=1}^\infty \frac{1}{k^2 \, 2^k}=\frac{\pi^2}{12}-\frac12 \log^2 2



\text{Li}_3\left(\frac12\right)=\sum_{k=1}^\infty \frac{1}{k^3 \, 2^k}=\frac78 \zeta(3)-\frac{\pi^2}{12} \log 2+\frac16 \log^3 2



\text{Li}_2\left(\frac{1}{\phi^2}\right)=\sum_{k=1}^\infty \frac{1}{k^2 \, \phi^{2k}}=\frac{\pi^2}{15}-\log^2 \phi


\text{Li}_2\left(\frac{1}{\phi}\right)=\sum_{k=1}^\infty \frac{1}{k^2 \, \phi^k}=\frac{\pi^2}{10}-\log^2 \phi


\text{Li}_2\left(-\frac{1}{\phi}\right)=\sum_{k=1}^\infty \frac{(-1)^k}{k^2 \, \phi^k}=-\frac{\pi^2}{15}+\frac12 \log^2 \phi



\text{Li}_3\left(\frac{1}{\phi^2}\right)=\sum_{k=1}^\infty \frac{1}{k^3 \, \phi^{2k}}=
\frac45 \zeta(3)+\frac23 \log^3\phi-\frac{2\pi^2}{15}\log\phi



[Bearbeiten] Fourier-Reihen


\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 -1<\varrho<1\; ,\; \varphi\in\Bbb{R}


\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\Bbb{R}



\sum_{k=1}^\infty \frac{(\pm 1)^k\, \cos(k\varphi)}{k}
=-\log\left(2\;\begin{matrix}\sin \\ \cos\end{matrix} \left(\frac{\varphi}{2}\right)\right) \qquad 0<\varphi<\pi


\sum_{k=1}^\infty \frac{(\pm 1)^k\, \sin(k\varphi)}{k}=\frac{\pi}{4}\pm \frac{\pi}{4}\mp\frac{\varphi}{2} \qquad 0<\varphi<\pi



\sum_{k=0}^\infty \frac{\cos (2k+1)\varphi}{2k+1}=-\frac12 \log\left( \tan \frac{\varphi}{2}\right) \qquad 0<\varphi<\pi



\sum_{k=0}^\infty \frac{\sin(2k+1)\varphi}{2k+1}=\frac{\pi}{4}\, \text{sgn}(\sin \varphi)


\sum_{k=0}^\infty \frac{(-1)^k\, \cos(2k+1)\varphi}{2k+1}=\frac{\pi}{4}\, \text{sgn}(\cos \varphi)


\sum_{k=0}^\infty \frac{(-1)^k\, \sin(2k+1)\varphi}{2k+1}=\frac12 \log\left(\tan\left(\frac{\pi}{4}+\frac{\varphi}{2}\right)\right) \qquad -\frac{\pi}{2}<\varphi<\frac{\pi}{2}



\sum_{k=0}^\infty \varrho^k\cos(k\varphi)=\frac{1-\varrho\cos\varphi}{1-2\varrho\cos\varphi+\varrho^2} \qquad |\varrho|<1\, , \, \varphi\in\Bbb{R}


\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\Bbb{R}



\sum_{k\in\Bbb{Z}} \frac{(-1)^k\, \cos kx}{k+\alpha}=\pi\, \frac{\cos \alpha x}{\sin \alpha\pi} \qquad -\pi<x<\pi \, , \, \alpha\notin\Bbb{Z}


\sum_{k\in\Bbb{Z}} \frac{(-1)^k\, \sin kx}{k+\alpha}=-\pi\, \frac{\sin \alpha x}{\sin \alpha\pi} \qquad -\pi<x<\pi \, , \, \alpha\notin\Bbb{Z}




\sum_{k\in\Bbb{Z}} \frac{(-1)^k\, \cos(kx)}{k^2+\alpha^2}=\frac{\pi}{\alpha}\, \frac{\cosh(\alpha x)}{\sinh(\alpha \pi)} \quad -\pi<x<\pi




\sum_{k\in\Bbb{Z}} \frac{\cos(2kx)}{k^4+\frac{\alpha^4}{4}}=\frac{2\pi}{\alpha^3} \begin{pmatrix}\sin\alpha x\, \cosh\alpha x-\cos\alpha x\,\sinh\alpha x \\ \\ +\frac{(\sinh\alpha\pi+\sin\alpha\pi)\cos\alpha x\,\cosh\alpha x-(\sinh\alpha\pi-\sin\alpha\pi)\sin\alpha x\,\sinh\alpha x}{\cosh\alpha\pi-\cos\alpha \pi}\end{pmatrix}\qquad 0<x<\pi \, , \, \alpha>0


\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\Bbb{R}



\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\Bbb{R}



[Bearbeiten] Poissonsche Summationsformel


\sum_{n\in\Bbb{Z}} f(n)=\sum_{n\in\Bbb{Z}} \hat{f}(n), wobei \hat{f}(n)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i nx}\, dx sein soll.



[Bearbeiten] Kummersche Reihe (Fourierreihenentwicklung der logΓ Funktion)


\sum_{n=1}^\infty \frac{\log n}{n\pi} \sin(2n\pi x)=\log\Gamma(x)-\frac12 \log 2\pi+\frac12 \log(2\,\sin\pi x)+
\left(\gamma+\log 2\pi\right)\left(x-\frac12\right) \qquad 0<x<1



[Bearbeiten] Zahlentheoretische Reihen


\sum_{(n,m)=1} \frac{1}{n\, m\, (n+m)}=2



\sum_{(n,m)=1} \frac{1}{n^s\, m^{s-1}\, (n+m)}=\frac{\zeta^2(s)}{2\, \zeta(2s)}



[Bearbeiten] Dirichlet Reihen


\sum_{n=1}^\infty \frac{\varphi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta(s)} \qquad \text{Re}(s)>2



\sum_{n=1}^\infty \frac{\psi(n)}{n^s}=\frac{\zeta(s)\, \zeta(s-1)}{\zeta(2s)} \qquad \text{Re}(s)>2



\sum_{n=1}^\infty \frac{\mu(n)}{n^s}=\frac{1}{\zeta(s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}=-\frac{\zeta'(s)}{\zeta(s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\tau^2(n)}{n^s}=\frac{\zeta^4(s)}{\zeta(2s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\tau(n^2)}{n^s}=\frac{\zeta^3(s)}{\zeta(2s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\mu^2(n)}{n^s}=\frac{\zeta(s)}{\zeta(2s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\lambda(n)}{n^s}=\frac{\zeta(2s)}{\zeta(s)} \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\tau(n)}{n^s}=\zeta^2(s) \qquad \text{Re}(s)>1



\sum_{n=1}^\infty \frac{\sigma(n)}{n^s}=\zeta(s)\, \zeta(s-1) \qquad \text{Re}(s)>2



\sum_{n=1}^\infty \frac{\sigma_a(n)}{n^s}=\zeta(s)\, \zeta(s-a) \qquad \text{Re}(s)>a+1



\sum_{n=1}^\infty \frac{\sigma_a(n)\, \sigma_b(n)}{n^s}=\frac{\zeta(s)\, \zeta(s-a)\, \zeta(s-b)\, \zeta(s-a-b)}{\zeta(2s-a-b)}



[Bearbeiten] Summatorische Funktionen


\sum_{d|n} \varphi(d)=n


\sum_{d|n} \frac{\mu(d)}{d}=\frac{\varphi(n)}{n}


\sum_{d|n} \lambda(d)=\left\{\begin{matrix} 1 & , & n \, \text{Quadratzahl} \\ 0 & , & \text{sonst} \end{matrix}\right.


\sum_{d|n} |\mu(d)|=2^{\omega(n)}\,


[Bearbeiten] Eisenstein Reihen


\frac{1}{2\zeta(2k)}\,{\sum_{n,m\in\Bbb{Z}}}' \frac{1}{(mz+n)^{2k}}=1-\frac{4k}{B_{2k}} \sum_{N=1}^\infty \sigma_{2k-1}(N)\, q^N \quad z\in\Bbb{H} \; , \; q=e^{2\pi i z}



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