Lemniscate 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:

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

The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that

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

so that

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

where Β denotes the beta function.

Gauss's constant should not be confused with the Gaussian gravitational constant.

Relations to other constants

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

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

Alternatively,

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

and since π and Γ(⁠1/4⁠) are algebraically independent with the Gauss's constant, Gauss's constant is transcendental.

Lemniscate constants

Gauss's constant may be used in the definition of the lemniscate constants, the first of which is:

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

and the second constant:

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

which arise in finding the arc length of a lemniscate.

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)

Record progression

Several world record attempts have been made to calculate the most digits of Gauss' constant or one of the lemniscate constants. Usually, the arclength of a lemniscate of radius = 1, or twice the first lemniscate constant, is calculated. Here is a chart for twice the first Lemniscate constant.

Date	Name	Number of digits
April 3, 2016	Ron Watkins	200 billion
Feb 9, 2016	Peter Trueb	190 billion
Dec 21, 2015	Ron Watkins	130 billion
Nov 14, 2015	Ron Watkins	125 billion
Oct 12, 2015	Ethan Gallagher	120 billion
July 5, 2015	Ron Watkins	100 billion
June 13, 2015	Andreas Stiller	80 billion
April 12, 2015	BenHadad	55 billion
March 19, 2015	Andreas Stiller	40 billion
February 9, 2015	Randy Ready	15 billion

See also

• Lemniscatic elliptic function

References

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