Gauss's constant

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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[edit]

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.

Lemniscate constants[edit]

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:


which arise in finding the arc length of a lemniscate.

Other formulas[edit]

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)

See also[edit]