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:
The constant is named after Carl Friedrich Gauss, who on May 30, 1799 discovered that
so that
where Β denotes the beta function.
Gauss's constant should not be confused with the Gaussian gravitational constant.
Relations to other constants
Gauss's constant may be used to express the gamma function at argument 1/4:
Alternatively,
and since π and Γ(1/4) are algebraically independent with the Gauss's constant, 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:
and the second constant:
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
as well as the rapidly converging series
The constant is also given by the infinite product
It appears in the evaluation of the integrals
Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...]. (sequence A053002 in the OEIS)
Record progression
Several world record attempts have been made to calculate the most digits of Gauss' constant or one of the lemniscate constants. Usually, the arclength of a lemniscate of radius = 1, or twice the first lemniscate constant, is calculated. Here is a chart for twice the first Lemniscate constant.[1]
Date | Name | Number of digits |
---|---|---|
April 3, 2016 | Ron Watkins | 200 billion |
Feb 9, 2016 | Peter Trueb | 190 billion |
Dec 21, 2015 | Ron Watkins | 130 billion |
Nov 14, 2015 | Ron Watkins | 125 billion |
Oct 12, 2015 | Ethan Gallagher | 120 billion |
July 5, 2015 | Ron Watkins | 100 billion |
June 13, 2015 | Andreas Stiller | 80 billion |
April 12, 2015 | BenHadad | 55 billion |
March 19, 2015 | Andreas Stiller | 40 billion |
February 9, 2015 | Randy Ready | 15 billion |
See also
References
- ↑ "Records set by y-cruncher". numberworld.org. Retrieved 3 December 2015.
- Sequences A014549 and A053002 in OEIS