# Formelsammlung Mathematik: Endliche Produkte

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##### 1
${\displaystyle \prod _{k=0}^{n-1}\Gamma \left(z+{\frac {k}{n}}\right)={\sqrt {2\pi }}^{\,n-1}\,n^{{\frac {1}{2}}-nz}\,\Gamma (nz)}$

##### 2
${\displaystyle \prod _{k=1}^{n-1}\Gamma \left({\frac {k}{n}}\right)={\frac {(2\pi )^{\frac {n-1}{2}}}{\sqrt {n}}}}$

##### 3
${\displaystyle \prod _{k=1}^{n-1}\left(1-\xi ^{k}\right)=n\qquad \xi =e^{\frac {2\pi i}{n}}}$

##### 4
${\displaystyle \alpha ^{2n}-2\alpha ^{n}\beta ^{n}\cos(n\theta )+\beta ^{2n}=\prod _{k=0}^{n-1}\left(\alpha ^{2}-2\alpha \beta \,\cos \left(\theta +{\frac {2\pi k}{n}}\right)+\beta ^{2}\right)}$

##### 5
${\displaystyle \prod _{k=0}^{n-1}\left(1+z^{2^{k}}\right)={\frac {1-z^{2^{n}}}{1-z}}\qquad z\neq 1}$

##### 6
${\displaystyle \prod _{k=1}^{n}{\frac {2}{z^{1/2^{k}}+z^{-1/2^{k}}}}=2^{n}\,{\frac {z^{1/2^{n}}-z^{-1/2^{n}}}{z-z^{-1}}}\qquad z\neq \pm 1}$

##### 7
${\displaystyle \prod _{k=0}^{n-1}\sin \left(z+{\frac {k\pi }{n}}\right)={\frac {\sin nz}{2^{n-1}}}}$

##### 8
${\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}}}$

##### 9
${\displaystyle \prod _{k=1}^{\lfloor {\frac {n-1}{2}}\rfloor }\tan \left({\frac {k\pi }{n}}\right)=\left\{{\begin{matrix}{\sqrt {n}}&,&n&{\text{ungerade}}\\\\1&,&n&{\text{gerade}}\end{matrix}}\right.}$

##### 10
${\displaystyle |\Gamma (n+ix)|^{2}=\prod _{k=0}^{n-1}(k^{2}+x^{2})\,{\frac {\pi x}{\sinh \pi x}}\qquad n\in \mathbb {Z} ^{\geq 0}\,,\,x\in \mathbb {R} }$

##### 11
${\displaystyle \left|\Gamma \left(n+{\frac {1}{2}}+ix\right)\right|^{2}=\prod _{k=0}^{n-1}\left(x^{2}+\left(k+{\frac {1}{2}}\right)^{2}\right)\,{\frac {\pi }{\cosh \pi x}}\qquad n\in \mathbb {Z} ^{\geq 0}\,,\,x\in \mathbb {R} }$