Catalanin vakio

Catalanin vakio G määritellään matematiikassa

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

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

$G=\int _{0}^{1}\int _{0}^{1}{\frac {1}{1+x^{2}y^{2}}}\,dx\,dy\!$

$G=-\int _{0}^{1}{\frac {\ln t}{1+t^{2}}}\,dt\!$

$G=\int _{0}^{\pi /4}{\frac {t}{\sin t\cos t}}\;dt\!$

$G={\frac {1}{4}}\int _{-\pi /2}^{\pi /2}{\frac {t}{\sin t}}\;dt\!$

$G=\int _{0}^{\pi /4}\ln(\cot(t))\,dt\!$

$G={\tfrac {1}{2}}\int _{0}^{\infty }{\frac {t}{\cosh t}}\,dt\;$

$G=\int _{0}^{\infty }\arctan(e^{-t})\,dt\!$

$G=\int _{0}^{1}{\frac {\arctan t}{t}}\,dt.\!$

$G={\frac {1}{2}}\int _{0}^{1}\mathrm {K} (t)\,dt\!$

missä K(t) on täydellinen elliptinen integraali.

Äärettömiä sarjoja

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

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

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

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