Formelsammlung Mathematik: Grenzwerte

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1

$\underset{|\xi |\to \mathrm{\infty }}{\lim }{(1+\frac{z}{\xi })}^{\xi }=e^z\mathrm{\forall }z\in ℂ$

2

$\underset{n\to \mathrm{\infty }}{\lim }\frac{n!n^z}{z(z+1)\cdots (z+n)}=\mathrm{\Gamma }(z)\mathrm{\forall }z\notin {\mathbb{Z}}_{\le 0}$

3

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n^y}\prod _{k=0}^n(1+\frac{y}{k+x})=\frac{\mathrm{\Gamma }(x)}{\mathrm{\Gamma }(x+y)}$

4

$\underset{\epsilon \to 0^{+}}{\lim }(\left|\mathrm{\Gamma }(-n+\epsilon e^{i{\varrho }_1})\right|-\left|\mathrm{\Gamma }(-n+\epsilon e^{i{\varrho }_2})\right|)=\frac{\psi (n+1)}{\mathrm{\Gamma }(n+1)}(\cos {\varrho }_1-\cos {\varrho }_2)$

$f$	$\underset{z_1,z_2\to 0}{\lim }\frac{z_{1}f(z_1)-z_{2}f(z_2)}{z_1-z_2}$
$\mathrm{\Gamma }(-n+z)$	$(-1{)}^n\frac{\mathrm{\Psi }(n+1)}{\mathrm{\Gamma }(n+1)}$
$\mathrm{\Gamma }(-n+z)z$	$(-1{)}^n\frac{1}{\mathrm{\Gamma }(n+1)}$
$\mathrm{\Psi }(-n+z)$	$\mathrm{\Psi }(n+1)$
$\mathrm{\Psi }(-n+z)z$	$-1$
$\zeta (1+z)(1+z{)}^{\alpha }$	$\alpha +\gamma$

5

$\underset{n\to \mathrm{\infty }}{\lim }\frac{n!e^n}{n^n\sqrt{2\pi n}}=1$

6

Mit $z=\varrho e^{i\phi }$ gilt $\mathrm{\Gamma }(z)\sim \sqrt{\frac{2\pi }{z}}{\left(\frac{z}{e}\right)}^z\{\begin{array}{cc}1& -\pi <\phi <\pi \\ \frac{1}{e^{2\pi iz}-1}& \phi =\pi \end{array}$ für $\varrho \to \mathrm{\infty }$

7

$\underset{n\to \mathrm{\infty }}{\lim }\frac{n}{\sqrt[n]{n!}}=e$

8

$\underset{n\to \mathrm{\infty }}{\lim }\frac{\sqrt{n\pi }}{2^{2n}}(\genfrac{}{}{0ex}{}{2n}{n})=1$

9

$B(\alpha n,\beta n)\sim \sqrt{\frac{2\pi }{n}}\sqrt{\frac{\alpha +\beta }{\alpha \beta }}{\left(\frac{{\alpha }^{\alpha }{\beta }^{\beta }}{(\alpha +\beta {)}^{\alpha +\beta }}\right)}^n$

10

$\underset{n\to \mathrm{\infty }}{\lim }\sqrt[n]{\frac{(\alpha n+\beta n)!}{(\alpha n)!(\beta n)!}}=\frac{(\alpha +\beta {)}^{\alpha +\beta }}{{\alpha }^{\alpha }{\beta }^{\beta }}$

11

$\underset{y\to \mathrm{\infty }}{\lim }\frac{|\mathrm{\Gamma }(x+iy)|}{\sqrt{\frac{2\pi }{y}}{y}^x{e}^{-\frac{\pi }{2}y}}=1x\in ℝ$

12

$A:=\underset{n\to \mathrm{\infty }}{\lim }\frac{e^{\frac{n^2}{4}}\prod _{k=1}^n{k}^k}{n^{\frac{n^2}{2}+\frac{n}{2}+\frac{1}{12}}}=e^{\frac{1}{12}-{\zeta }^{\prime }(-1)}$

13

$\underset{n\to \mathrm{\infty }}{\lim }[\frac{(n+1{)}^{n+1}}{n^n}-\frac{n^n}{(n-1{)}^{n-1}}]=e$

14

$\underset{m\to \mathrm{\infty }}{\lim }(\sum _{k=1}^m\frac{(\log k{)}^n}{k}-\frac{(\log m{)}^{n+1}}{n+1})={\gamma }_n$

15

$\underset{n\to \mathrm{\infty }}{\lim }(H_n-\log n)=\gamma$

16

$\underset{n\to \mathrm{\infty }}{\lim }(\log n-\sum _{k=0}^n\frac{1}{k+x})=\psi (x)$

17

$\underset{z\to 1}{\lim }(\zeta (z)-\frac{1}{z-1})=\gamma$

18

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n!}\underset{n+a\sqrt{n}}{\overset{n+b\sqrt{n}}{\int }}\frac{x^n}{e^x}dx=\frac{1}{\sqrt{2\pi }}\underset{a}{\overset{b}{\int }}{e}^{-\frac{x^2}{2}}dxa,b\in ℂ$