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.
Gauss and others use the equivalent of
which is the lemniscate constant, prominent in the theory of lemniscatic functions.
However, John Todd uses a different terminology – in his article, the numbers and are called the lemniscate constants,
the first of which is
and the second constant:
They arise in finding the arc length of a lemniscate of Bernoulli. and were proven transcendental by Theodor Schneider in 1937 and 1941, respectively.
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)