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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 in the definition of the lemniscate constants, the first of which is:
and the second constant:
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
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.
|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|
- "Records set by y-cruncher". numberworld.org. Retrieved 3 December 2015.