Gauss's constant
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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:
The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that
so that
where β denotes the beta function.
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[edit] Relations to other constants
Gauss's constant may be used as a closed-form expression for the Gamma function at argument 1/4:
and since π and Γ(1/4) are algebraically independent, Gauss's constant is transcendental.
[edit] Lemniscate constants
Gauss's constant may be used in the definition of the lemniscate constants, the first of which is:
and the second constant:
which arise in finding the arc length of a lemniscate.
[edit] Other formulas
A formula for G in terms of Jacobi theta functions is given by:
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.](../../../math/f/9/0/f90850c37d296a9232a0e56ecd54d446.png)
The constant is also given by the infinite product
Gauss's constant has continued fraction [0, 1, 5, 21, 3, 4, 14...].
[edit] References
- Eric W. Weisstein. "Gauss's Constant." From MathWorld--A Wolfram Web Resource.
- Sequences A014549 and A053002 in OEIS
