Gauss's constant

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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:

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

The constant is named after Carl Friedrich Gauss, who in 1799^[1] discovered that

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

so that

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

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:

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

Alternatively,

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

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^[2][3] use the equivalent of

${\displaystyle \varpi =\pi G}$

which is the lemniscate constant.

However, John Todd uses a different terminology, defining two "lemniscate constants" ${\displaystyle A}$ and ${\displaystyle B}$:^[4]

${\displaystyle {\begin{aligned}A&={\tfrac {1}{2}}\pi G={\tfrac {1}{2}}\varpi ={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{4}},{\tfrac {1}{2}}{\bigr )},\\[3mu]B&={\frac {1}{2G}}={\tfrac {1}{4}}\mathrm {B} {\bigl (}{\tfrac {1}{2}},{\tfrac {3}{4}}{\bigr )}.\end{aligned}}}$

They arise in finding the arc length of a lemniscate of Bernoulli. ${\displaystyle A}$ and ${\displaystyle B}$ were proven transcendental by Theodor Schneider in 1937 and 1941, respectively.^[4]

Other formulas[]

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

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

as well as the rapidly converging series

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

The constant is also given by the infinite product

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

An analog of the Wallis product is:^[5]

${\displaystyle G=\prod _{n=1}^{\infty }\left({\frac {4n-1}{4n}}\cdot {\frac {4n+2}{4n+1}}\right)={\biggl (}{\frac {3}{4}}\cdot {\frac {6}{5}}{\biggr )}{\biggl (}{\frac {7}{8}}\cdot {\frac {10}{9}}{\biggr )}{\biggl (}{\frac {11}{12}}\cdot {\frac {14}{13}}{\biggr )}\cdots }$

It appears in the evaluation of the integrals

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

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

Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...]. (sequence A053002 in the OEIS)

See also[]

• Lemniscatic elliptic function

References[]

• Weisstein, Eric W. "Gauss's Constant". MathWorld.
• Sequences A014549 and A053002 in OEIS

External links[]

• "Gauss's constant and where it occurs". www.johndcook.com. 2021-10-17.