Formelsammlung Mathematik: Unendliche Produkte

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1

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

2

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

3

$\prod _{n=1}^{\mathrm{\infty }}{(1-q^n)}^3=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n(2n+1)q^{\frac{n(n+1)}{2}}|q|<1$

4

$\prod _{n=1}^{\mathrm{\infty }}(1-q^n)=\sum _{n\in \mathbb{Z}}(-1{)}^n{q}^{\frac{n(3n+1)}{2}}|q|<1$

5

$\prod _{k=1}^{\mathrm{\infty }}(1+z^k)=\prod _{k=1}^{\mathrm{\infty }}(1-z^{2k-1}{)}^{-1}|z|<1$

6

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

7

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

8.1

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

8.2

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

9

$\sin \pi z=\pi z\prod _{k=1}^{\mathrm{\infty }}(1-\frac{z^2}{k^2})z\in ℂ$

10

$\sinh \pi z=\pi z\prod _{k=1}^{\mathrm{\infty }}(1+\frac{z^2}{k^2})z\in ℂ$

11

$\cos \pi z=\prod _{k=1}^{\mathrm{\infty }}(1-\frac{z^2}{{(k-\frac{1}{2})}^2})z\in ℂ$

12

$\cosh \pi z=\prod _{k=1}^{\mathrm{\infty }}(1+\frac{z^2}{{(k-\frac{1}{2})}^2})z\in ℂ$

13

$\prod _{k=1}^{\mathrm{\infty }}(1-\frac{z^n}{k^n})={[\prod _{k=0}^{n-1}\mathrm{\Gamma }(1-{\xi }^{k}z)]}^{-1}\xi =e^{\frac{2\pi i}{n}}$

14

$\mathrm{\Gamma }(z)=\frac{1}{z}\prod _{k=1}^{\mathrm{\infty }}\frac{{(1+\frac{1}{k})}^z}{1+\frac{z}{k}}z\notin {\mathbb{Z}}_{\le 0}$

15

$\mathrm{\Gamma }(z)=\frac{1}{z}{e}^{-\gamma z}\prod _{k=1}^{\mathrm{\infty }}\frac{e^{\frac{z}{k}}}{1+\frac{z}{k}}z\notin {\mathbb{Z}}_{\le 0}$

16

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

17

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

18

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

19

$\prod _{k=0}^{\mathrm{\infty }}(1+\frac{(-1{)}^k}{ak+b})=\frac{\mathrm{\Gamma }\left(\frac{b}{2a}\right)\mathrm{\Gamma }\left(\frac{a+b}{2a}\right)}{\mathrm{\Gamma }\left(\frac{b+1}{2a}\right)\mathrm{\Gamma }\left(\frac{a+b-1}{2a}\right)}a\in ℂ\setminus \{0\},\frac{b}{a}\in ℂ\setminus {\mathbb{Z}}^{\le 0}$

20

$\prod _{k=1}^{\mathrm{\infty }}\frac{1-e^{-2\pi kz}}{1-e^{-2\pi k/z}}=\frac{e^{\frac{\pi }{12}(z-\frac{1}{z})}}{\sqrt{z}}\text{Re}(z)>0$

21

$\prod _{n=0}^{\mathrm{\infty }}{(\prod _{k=0}^n(k+1{)}^{(-1{)}^{k+1}(\genfrac{}{}{0ex}{}{n}{k})})}^{\frac{n\cdot (n+1)}{2^{n+3}}}=e^{\frac{7\zeta (3)}{24\zeta (2)}}$