Gauss's constant

In mathematics, Gauss' 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}(1, \sqrt{2})} = 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}B( \tfrac{1}{4}, \tfrac{1}{2})

where B denotes the beta function.

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

Relations to other constants

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

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

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

Gauss' 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.

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

G = \vartheta_{01}^2(e^{-\pi})

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^{\pi/2}\sqrt{\sin(x)}dx=\int_0^{\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 OEIS)