Lemniscate constant

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 $π$ for the circle. Equivalently, the perimeter of the lemniscate $(x^{2}+y^{2})^{2}=x^{2}-y^{2}$ is $2 ϖ$ . 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 $ϖ / π \approx 0.8346268$ .

John Todd named two more lemniscate constants, the first lemniscate constant $A = ϖ /2 \approx 1.3110287771$ and the second lemniscate constant $B = π /(2 ϖ) \approx 0.5990701173$ .

Sometimes the quantities $2 ϖ$ or $A$ are referred to as the lemniscate constant.

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