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 in 1799 discovered that
where Β denotes the beta function.
Relations to other constants
Gauss's constant may be used to express the gamma function at argument 1/4:
and since π and Γ(1/4) are algebraically independent, Gauss's constant is transcendental.
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. Both constants were proven to be transcendental.
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)