Formelsammlung Mathematik: Endliche Produkte

Zurück zur Formelsammlung Mathematik

1

$\prod _{k=0}^{n-1}\mathrm{\Gamma }(z+\frac{k}{n})={\sqrt{2\pi }}^{n-1}{n}^{\frac{1}{2}-nz}\mathrm{\Gamma }(nz)$

2

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

3

$\prod _{k=1}^{n-1}(1-{\xi }^k)=n\xi =e^{\frac{2\pi i}{n}}$

4

${\alpha }^{2n}-2{\alpha }^n{\beta }^n\cos (n\theta )+{\beta }^{2n}=\prod _{k=0}^{n-1}({\alpha }^2-2\alpha \beta \cos (\theta +\frac{2\pi k}{n})+{\beta }^2)$

5

$\prod _{k=0}^{n-1}(1+z^{2^k})=\frac{1-z^{2^n}}{1-z}z\ne 1$

6

$\prod _{k=1}^n\frac{2}{z^{1/2^k}+z^{-1/2^k}}=2^n\frac{z^{1/2^n}-z^{-1/2^n}}{z-z^{-1}}z\ne \pm 1$

7

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

8

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

9

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

10

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

11

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