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# Formelsammlung Mathematik: Unendliche Produkte

Zurück zur Formelsammlung Mathematik

##### 1
${\displaystyle \prod _{k=1}^{\infty }{\frac {4k^{2}}{4k^{2}-1}}={\frac {\pi }{2}}}$

##### 2
${\displaystyle \prod _{n=1}^{\infty }\left(\left(1-x^{2n}\right)\left(1+x^{2n-1}z^{2}\right)\left(1+{\frac {x^{2n-1}}{z^{2}}}\right)\right)=\sum _{n\in \mathbb {Z} }x^{n^{2}}\,z^{2n}\qquad |x|<1\;,\;z\neq 0}$

##### 3
${\displaystyle \prod _{n=1}^{\infty }\left(1-q^{n}\right)^{3}=\sum _{n=0}^{\infty }(-1)^{n}\,(2n+1)\,q^{\frac {n(n+1)}{2}}\qquad |q|<1}$

##### 4
${\displaystyle \prod _{n=1}^{\infty }\left(1-q^{n}\right)=\sum _{n\in \mathbb {Z} }(-1)^{n}\,q^{\frac {n\,(3n+1)}{2}}\qquad |q|<1}$

##### 5
${\displaystyle \prod _{k=1}^{\infty }(1+z^{k})=\prod _{k=1}^{\infty }(1-z^{2k-1})^{-1}\qquad |z|<1}$

##### 6
${\displaystyle {\frac {e}{2}}=\left({\frac {4}{3}}\right)^{\frac {1}{2}}\cdot \left({\frac {6\cdot 8}{5\cdot 7}}\right)^{\frac {1}{4}}\cdot \left({\frac {10\cdot 12\cdot 14\cdot 16}{9\cdot 11\cdot 13\cdot 15}}\right)^{\frac {1}{8}}\cdots }$

##### 7
${\displaystyle {\frac {e}{2}}=\left({\frac {2}{1}}\right)^{\frac {1}{2}}\cdot \left({\frac {2\cdot 4}{3\cdot 3}}\right)^{\frac {1}{4}}\cdot \left({\frac {4\cdot 6\cdot 6\cdot 8}{5\cdot 5\cdot 7\cdot 7}}\right)^{\frac {1}{8}}\cdots }$

##### 8.1
${\displaystyle {\text{sinc}}\,z=\prod _{k=1}^{\infty }\cos \left({\frac {z}{2^{k}}}\right)={\sqrt {{\frac {1}{2}}+{\frac {\cos z}{2}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {\cos z}{2}}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {\cos z}{2}}}}}}}}\cdots }$

##### 8.2
${\displaystyle {\frac {2}{\pi }}={\sqrt {\frac {1}{2}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}\cdot {\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {\frac {1}{2}}}}}}}\cdots ={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots }$

##### 9
${\displaystyle \sin \pi z=\pi z\,\prod _{k=1}^{\infty }\left(1-{\frac {z^{2}}{k^{2}}}\right)\qquad z\in \mathbb {C} }$

##### 10
${\displaystyle \sinh \pi z=\pi z\,\prod _{k=1}^{\infty }\left(1+{\frac {z^{2}}{k^{2}}}\right)\qquad z\in \mathbb {C} }$

##### 11
${\displaystyle \cos \pi z=\prod _{k=1}^{\infty }\left(1-{\frac {z^{2}}{\left(k-{\frac {1}{2}}\right)^{2}}}\right)\qquad z\in \mathbb {C} }$

##### 12
${\displaystyle \cosh \pi z=\prod _{k=1}^{\infty }\left(1+{\frac {z^{2}}{\left(k-{\frac {1}{2}}\right)^{2}}}\right)\qquad z\in \mathbb {C} }$

##### 13
${\displaystyle \prod _{k=1}^{\infty }\left(1-{\frac {z^{n}}{k^{n}}}\right)=\left[\prod _{k=0}^{n-1}\Gamma (1-\xi ^{k}z)\right]^{-1}\qquad \xi =e^{\frac {2\pi i}{n}}}$

##### 14
${\displaystyle \Gamma (z)={\frac {1}{z}}\,\prod _{k=1}^{\infty }{\frac {\left(1+{\frac {1}{k}}\right)^{z}}{1+{\frac {z}{k}}}}\qquad z\notin \mathbb {Z} _{\leq 0}}$

##### 15
${\displaystyle \Gamma (z)={\frac {1}{z}}\,e^{-\gamma z}\,\prod _{k=1}^{\infty }{\frac {e^{\frac {z}{k}}}{1+{\frac {z}{k}}}}\qquad z\notin \mathbb {Z} _{\leq 0}}$

##### 16
${\displaystyle \prod _{k=1}^{\infty }\left[{\frac {1}{\sqrt {\pi }}}\,\Gamma \left({\frac {1}{2}}+{\frac {z}{2^{k}}}\right)\right]=2^{-2z}\,\Gamma (1+z)\qquad z\notin \mathbb {Z} _{<0}}$

##### 17
${\displaystyle \prod _{k=0}^{\infty }\left[\left(1+{\frac {y}{k+x}}\right)e^{-{\frac {y}{k+x}}}\right]={\frac {\Gamma (x)\,e^{y\,\psi (x)}}{\Gamma (x+y)}}}$

##### 18
${\displaystyle {\frac {\Gamma (x)}{|\Gamma (x+iy)|}}=\prod _{k=0}^{\infty }{\sqrt {1+\left({\frac {y}{x+k}}\right)^{2}}}}$

##### 19
${\displaystyle \prod _{k=0}^{\infty }\left(1+{\frac {(-1)^{k}}{ak+b}}\right)={\frac {\Gamma \!\left({\frac {b}{2a}}\right)\,\Gamma \!\left({\frac {a+b}{2a}}\right)}{\Gamma \!\left({\frac {b+1}{2a}}\right)\,\Gamma \!\left({\frac {a+b-1}{2a}}\right)}}\qquad a\in \mathbb {C} \setminus \{0\}\quad ,\quad {\frac {b}{a}}\in \mathbb {C} \setminus \mathbb {Z} ^{\leq 0}}$

##### 20
${\displaystyle \prod _{k=1}^{\infty }{\frac {1-e^{-2\pi kz}}{1-e^{-2\pi k/z}}}={\frac {e^{{\frac {\pi }{12}}\left(z-{\frac {1}{z}}\right)}}{\sqrt {z}}}\qquad {\text{Re}}(z)>0}$

##### 21
${\displaystyle \prod _{n=0}^{\infty }\left(\prod _{k=0}^{n}(k+1)^{(-1)^{k+1}\,{n \choose k}}\right)^{\frac {n\cdot (n+1)}{2^{n+3}}}=e^{\frac {7\,\zeta (3)}{24\,\zeta (2)}}}$