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| In [[mathematics]], '''Gauss' constant''', denoted by ''G'', is defined as the [[Multiplicative inverse|reciprocal]] of the [[arithmetic-geometric mean]] of [[1 (number)|1]] and the [[square root of 2]]:
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| : <math> G = \frac{1}{\mathrm{agm}(1, \sqrt{2})} = 0.8346268\dots.</math>
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| The [[mathematical constant|constant]] is named after [[Carl Friedrich Gauss]], who on May 30, 1799 discovered that
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| : <math> G = \frac{2}{\pi}\int_0^1\frac{dx}{\sqrt{1 - x^4}} </math> | |
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| so that
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| : <math> G = \frac{1}{2\pi}B( \tfrac{1}{4}, \tfrac{1}{2})</math>
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| where ''B'' denotes the [[beta function]].
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| Gauss' constant should not be confused with the [[Gaussian gravitational constant]].
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| ==Relations to other constants== | |
| Gauss' constant may be used to express the [[Gamma function]] at [[particular values of the Gamma function|argument 1/4]]:
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| : <math> \Gamma( \tfrac{1}{4}) = \sqrt{ 2G \sqrt{ 2\pi^3 } } </math>
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| and since π and Γ(1/4) are [[algebraically independent]] with Γ(1/4) irrational, Gauss' constant is [[transcendental number|transcendental]].
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| ===Lemniscate constants===
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| Gauss' constant may be used in the definition of the lemniscate constants, the first of which is:
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| : <math> L_1\;=\;\pi G </math>
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| and the second constant: | |
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| : <math> L_2\,\,=\,\,\frac{1}{2G} </math>
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| which arise in finding the [[arc length]] of a [[lemniscate of Bernoulli|lemniscate]].
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| ==Other formulas==
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| A formula for ''G'' in terms of [[theta function|Jacobi theta function]]s is given by
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| : <math>G = \vartheta_{01}^2(e^{-\pi}) </math>
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| as well as the rapidly converging series
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| : <math>G = \sqrt[4]{32}e^{-\frac{\pi}{3}}\left (\sum_{n = -\infty}^\infty (-1)^n e^{-2n\pi(3n+1)} \right )^2.</math>
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| The constant is also given by the [[infinite product]]
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| :<math>G = \prod_{m = 1}^\infty \tanh^2 \left( \frac{\pi m}{2}\right).</math>
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| It appears in the evaluation of the integrals
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| : <math> {\frac{1}{G}} = \int_0^{\pi/2}\sqrt{\sin(x)}dx=\int_0^{\pi/2}\sqrt{\cos(x)}dx </math>
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| : <math> G = \int_0^{\infty}{\frac{dx}{\sqrt{\cosh(\pi x)}}} </math>
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| Gauss' constant as a [[continued fraction]] is [0, 1, 5, 21, 3, 4, 14, ...].
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| ==See also==
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| *[[Lemniscatic elliptic function]]
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| ==References==
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| *{{mathworld|urlname=GausssConstant|title=Gauss's Constant}}
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| * Sequences A014549 and A053002 in [[OEIS]]
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| [[Category:Mathematical constants]]
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| [[Category:Transcendental numbers]]
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