Concept
In mathematics, the lemniscate constant π is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli's lemniscate to its diameter, analogous to the definition of pi for the circle. Equivalently, the perimeter of the lemniscate is 2π. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755. The symbol π is a cursive variant of π; see Pi § Variant pi. Gauss's constant, denoted by G, is equal to π /pi ≈ 0.8346268. John Todd named two more lemniscate constants, the first lemniscate constant A = π/2 ≈ 1.3110287771 and the second lemniscate constant B = pi/(2π) ≈ 0.5990701173. Sometimes the quantities 2π or A are referred to as the lemniscate constant. Gauss's constant is named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as . By 1799, Gauss had two proofs of the theorem that where is the lemniscate constant. The lemniscate constant and first lemniscate constant were proven transcendental by Theodor Schneider in 1937 and the second lemniscate constant and Gauss's constant were proven transcendental by Theodor Schneider in 1941. In 1975, Gregory Chudnovsky proved that the set is algebraically independent over , which implies that and are algebraically independent as well. But the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . In fact, Usually, is defined by the first equality below. where K is the complete elliptic integral of the first kind with modulus k, Β is the beta function, Γ is the gamma function and ζ is the Riemann zeta function. The lemniscate constant can also be computed by the arithmetic–geometric mean , Moreover, which is analogous to where is the Dirichlet beta function and is the Riemann zeta function. Gauss's constant is typically defined as the reciprocal of the arithmetic–geometric mean of 1 and the square root of 2, after his calculation of published in 1800: Gauss's constant is equal to where Β denotes the beta function.