Gauss\'s constant - Citizendia

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the arithmetic-geometric mean (AGM of two positive Real numbers x and y is defined as follows First compute the Arithmetic Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity The square root of 2, also known as Pythagoras' Constant, often denoted by \sqrt{2} or √2 A mathematical constant is a number usually a Real number, that arises naturally in Mathematics. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Mathematics, the beta function, also called the Euler integral of the first kind is a Special function defined by

1 Relations to other constants
- 1.1 Lemniscate constants
2 Other formulas
3 References

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. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function The Gamma function is an important Special function in Mathematics. In Abstract algebra, a Subset S of a field L is algebraically independent over a subfield K if the elements In Mathematics, a transcendental number is a Complex number that is not algebraic, that is not a solution of a non-zero Polynomial equation

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. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Algebraic geometry, the word lemniscate refers to any of several figure-eight or ∞ shaped curves of which the best known is the Lemniscate of Bernoulli

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

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

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, . In Mathematics, theta functions are Special functions of Several complex variables. In Mathematics, for a Sequence of numbers a 1 a 2 a 3. the infinite product In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} . . ].

References

Eric W. Weisstein, Gauss's Constant at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Sequences A014549 and A053002 in OEIS

The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
network: | |

Contents

Relations to other constants

Lemniscate constants

Other formulas

References