Mathematik-Online-Lexikon: Spezielle komplexe Reihen und Produkte

$\displaystyle \frac{\pi^2}{\sin^2(\pi z)}$	$\displaystyle = \sum_{n=-\infty}^\infty \frac{1}{(z-n)^2}$
$\displaystyle \pi\cot(\pi z)$	$\displaystyle = \sum_{n=-\infty}^\infty \frac{1}{z-n}= \frac{1}{z}+\sum_{n=1}^\infty \frac{2z}{z^2-n^2}$
$\displaystyle \frac{\pi}{\sin(\pi z)}$	$\displaystyle = \sum_{n=-\infty}^\infty \frac{(-1)^n}{z-n} = \frac{1}{z}+\sum_{n=1}^\infty (-1)^n\frac{2z}{z^2-n^2}$
$\displaystyle \sin(\pi z)$	$\displaystyle = \pi z \prod_{n=1}^\infty\left(1-\frac{z^2}{n^2}\right)$
$\displaystyle \Gamma(z)$	$\displaystyle = \frac{e^{-\gamma z}}{z}\prod_{n=1}^\infty \left(1+\frac{z}{n}\right)^{-1}e^{z/n}$
$\displaystyle \frac{\Gamma^2(n+1)}{\Gamma(n+x\mathrm{i}+1)\Gamma(n-x\mathrm{i}+1)}$	$\displaystyle =\prod_{k=1}^\infty \left(1+\frac{x^2}{(n+k)^2}\right)$
$\displaystyle \frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n+1}{2}\right)} \frac{\cosh(\pi n\sqrt{3})-\cos(\pi n)}{2^{n+2}\pi^{3/2}n}$	$\displaystyle = \prod_{k=1}^\infty \left(1+\left(\frac{n}{k}\right)^3\right) \prod_{k=1}^\infty \left(1+3\left(\frac{n}{n+2k}\right)^2\right)$

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	Mathematik-Online-Lexikon:
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