# Gauss's constant

In mathematics, Gauss's constant, denoted by G, is defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2:

$G={\frac {1}{\operatorname {agm} \left(1,{\sqrt {2}}\right)}}=0.8346268\dots .$ The constant is named after Carl Friedrich Gauss, who in 1799 discovered that

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

$G={\frac {1}{2\pi }}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ where Β denotes the beta function.

## Relations to other constants

Gauss's constant may be used to express the gamma function at argument 1/4:

$\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}={\sqrt {2G{\sqrt {2\pi ^{3}}}}}$ Alternatively,

$G={\frac {\Gamma {\bigl (}{\tfrac {1}{4}}{\bigr )}{}^{2}}{2{\sqrt {2\pi ^{3}}}}}$ and since π and Γ(1/4) are algebraically independent, Gauss's constant is transcendental.

### Lemniscate constants

Gauss's constant may be used in the definition of the lemniscate constants.

Gauss and others use the equivalent of

$\varpi =\pi G$ which is the lemniscate constant, prominent in the theory of lemniscatic functions.

However, John Todd uses a different terminology – in his article, the numbers $A$ and $B$ are called the lemniscate constants, the first of which is

$A={\tfrac {1}{2}}\pi G={\tfrac {1}{2}}\varpi ={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )}$ and the second constant:

$B={\frac {1}{2G}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.$ They arise in finding the arc length of a lemniscate of Bernoulli. $A$ and $B$ were proven transcendental by Theodor Schneider in 1937 and 1941, respectively.

## Other formulas

A formula for G in terms of Jacobi theta functions is given by

$G=\vartheta _{01}^{2}\left(e^{-\pi }\right)$ as well as the rapidly converging series

$G={\sqrt[{4}]{32}}e^{-{\frac {\pi }{3}}}\left(\sum _{n=-\infty }^{\infty }(-1)^{n}e^{-2n\pi (3n+1)}\right)^{2}.$ The constant is also given by the infinite product

$G=\prod _{m=1}^{\infty }\tanh ^{2}\left({\frac {\pi m}{2}}\right).$ It appears in the evaluation of the integrals

${\frac {1}{G}}=\int _{0}^{\frac {\pi }{2}}{\sqrt {\sin(x)}}\,dx=\int _{0}^{\frac {\pi }{2}}{\sqrt {\cos(x)}}\,dx$ $G=\int _{0}^{\infty }{\frac {dx}{\sqrt {\cosh(\pi x)}}}$ Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...]. (sequence A053002 in the OEIS)