Catalanin vakio

Catalanin vakio G määritellään matematiikassa

$G=\beta (2)=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^2}=\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots$,

jossa $\beta$ on Dirichlet'n betafunktio.

Sen likiarvo on

$G\approx 0,915965594177219015054603514932$. [1]

Ei tiedetä, onko Catalanin vakio rationaalinen vai irrationaalinen.

Catalanin vakio on nimetty belgialaisen matemaatikon Eugène Charles Catalanin mukaan.

Integraaleja

$G=\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{1+x^2{y}^2}dxdy$

$G=-\underset{0}{\overset{1}{\int }}\frac{\ln t}{1+t^2}dt$

$G=\underset{0}{\overset{\pi /4}{\int }}\frac{t}{\sin t\cos t}dt$

$G=\frac{1}{4}\underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{t}{\sin t}dt$

$G=\underset{0}{\overset{\pi /4}{\int }}\ln (\cot (t))dt$

$G=\frac{1}{2}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t}{\cosh t}dt$

$G=\underset{0}{\overset{\mathrm{\infty }}{\int }}\arctan (e^{-t})dt$

$G=\underset{0}{\overset{1}{\int }}\frac{\arctan t}{t}dt.$

$G=\frac{1}{2}\underset{0}{\overset{1}{\int }}\mathrm{K}(t)dt$

missä K(t) on täydellinen elliptinen integraali.

Äärettömiä sarjoja

$G=\frac{1}{16}\sum _{n=1}^{\mathrm{\infty }}(n+1)\frac{3^n-1}{4^n}\zeta (n+2).$

$G=\frac{1}{64}\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}\cdot 2^{8n}\cdot (40n^2-24n+3)\cdot (2n){!}^3\cdot n{!}^2}{n^3\cdot (2n-1)\cdot (4n){!}^2}.$

$\begin{array}{cc}G& =3\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^{4n}}(-\frac{1}{2(8n+2{)}^2}+\frac{1}{2^2(8n+3{)}^2}-\frac{1}{2^3(8n+5{)}^2}+\frac{1}{2^3(8n+6{)}^2}-\frac{1}{2^4(8n+7{)}^2}+\frac{1}{2(8n+1{)}^2})\\ & -2\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^{12n}}(\frac{1}{2^4(8n+2{)}^2}+\frac{1}{2^6(8n+3{)}^2}-\frac{1}{2^9(8n+5{)}^2}-\frac{1}{2^{10}(8n+6{)}^2}-\frac{1}{2^{12}(8n+7{)}^2}+\frac{1}{2^3(8n+1{)}^2})\end{array}$

$G=\frac{1}{8}\pi \log (2+\sqrt{3})+\frac{3}{8}\sum _{n=0}^{\mathrm{\infty }}\frac{(n!{)}^2}{(2n)!(2n+1{)}^2}.$