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}\Beta\bigl( \tfrac14, \tfrac12\bigr)
where Β denotes the beta function.
Relations to other constants
Gauss's constant may be used to express the gamma function at argument :
\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  and Γ() 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.
However, John Todd uses a different terminology, defining two "lemniscate constants" A and B:
\begin{aligned} A &= \tfrac12\pi G = \tfrac12\varpi = \tfrac14 \Beta \bigl(\tfrac14,\tfrac12\bigr), \\[3mu] B &= \frac{1}{2G} =\tfrac14\Beta \bigl(\tfrac12,\tfrac34\bigr).
\end{aligned}
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).
An analog of the Wallis product is:
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
{\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, ...].
See also
Lemniscatic elliptic function
References
Sequences A014549 and A053002 in OEIS
External links
