where B denotes the beta function.
Gauss' constant should not be confused with the Gaussian gravitational constant.
Relations to other constants
Gauss' 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
Gauss' constant as a continued fraction is [0, 1, 5, 21, 3, 4, 14, ...].