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:

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

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

$G = \frac{2}{\pi}\beta(\begin{matrix} \frac{1}{4}\end{matrix}, \begin{matrix}\frac{1}{4}\end{matrix})$

where β denotes the beta function.

[edit] Relations to other constants

Gauss's constant may be used as a closed-form expression for the Gamma function at argument 1/4:

$\Gamma( \begin{matrix} \frac{1}{4} \end{matrix}) = \sqrt{ 2G \sqrt{ 2\pi^3 } }$

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

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.

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

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

Gauss's constant has continued fraction [0, 1, 5, 21, 3, 4, 14, ...].