# 高斯常數

${\displaystyle G={\frac {1}{\mathrm {agm} (1,{\sqrt {2}})}}=0.8346268\dots .}$

${\displaystyle G={\frac {2}{\pi }}\int _{0}^{1}{\frac {dx}{\sqrt {1-x^{4}}}}}$

${\displaystyle G={\frac {1}{2\pi }}B({\tfrac {1}{4}},{\tfrac {1}{2}})}$

## 和其他常數的關係

${\displaystyle \Gamma ({\tfrac {1}{4}})={\sqrt {2G{\sqrt {2\pi ^{3}}}}}}$

${\displaystyle G={\frac {[\Gamma ({\tfrac {1}{4}})]^{2}}{2{\sqrt {2\pi ^{3}}}}}}$

### Lemniscate常數

${\displaystyle L_{1}\;=\;\pi G}$

${\displaystyle L_{2}\,\,=\,\,{\frac {1}{2G}}}$

## 其他公式

${\displaystyle G=\vartheta _{01}^{2}(e^{-\pi })}$

${\displaystyle G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.}$

${\displaystyle G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).}$

${\displaystyle {\frac {1}{G}}=\int _{0}^{\pi /2}{\sqrt {\sin(x)}}dx=\int _{0}^{\pi /2}{\sqrt {\cos(x)}}dx}$
${\displaystyle G=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}}$

## 參考資料

• Sequences A014549 and A053002 in OEIS