Gauss's constant

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})} = 0.8346268dots

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

: G = frac{2}{pi}int_0^1frac{dx}{sqrt{1 - x^4

so that

: G = frac{2}{pi}eta(egin{matrix} frac{1}{4}end{matrix}, egin{matrix}frac{1}{4}end{matrix})

where β denotes the beta function.

Relations to other constants

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

: Gamma( egin{matrix} frac{1}{4} end{matrix}) = sqrt{ 2G sqrt{ 2pi^3 } }

and since π and Γ(1/4) are algebraically independent, 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(e^{-pi})

as well as the rapidly converging series

: G = sqrt [4] {32}e^{-frac{pi}{3left (sum_{n = -infty}^{infty} (-1)^n e^{-2npi(3n+1)} ight )^2.

The constant is also given by the infinite product

:G = prod_{m = 1}^infty anh^2 left( frac{pi m}{2} ight).

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

References

*mathworld|urlname=GausssConstant|title=Gauss's Constant
* Sequences A014549 and A053002 in OEIS


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