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.

[2] It also appears in evaluation of the gamma and beta function at certain rational values.

The symbol ϖ is a cursive variant of π known as variant pi represented in Unicode by the character U+03D6 ϖ GREEK PI SYMBOL.

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

[3][4] As of 2024 over 1.2 trillion digits of this constant have been calculated.

[5] Gauss's constant, denoted by G, is equal to ϖ /π ≈ 0.8346268[6] and named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as

were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's second lemniscate constant

were proven transcendental by Theodor Schneider in 1941.

(where the prime denotes the derivative with respect to the second variable) is not algebraically independent over

[16] In 1996, Yuri Nesterenko proved that the set

is defined by the first equality below, but it has many equivalent forms:[18]

The lemniscate constant can also be computed by the arithmetic–geometric mean

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

John Todd's lemniscate constants may be given in terms of the beta function B:

[20] Analogously to the Leibniz formula for π,

) and also tells us that the BSD conjecture is true for the above

Viète's formula for π can be written:

An infinite series discovered by Gauss is:[34]

and several similar formulas for π can be developed using trigonometric angle sum identities, e.g. Euler's formula

Analogous formulas can be developed for ϖ, including the following found by Gauss:

[35] The lemniscate constant can be rapidly computed by the series[36][37] where

⁠ (see Lemniscate elliptic functions § Hurwitz numbers for a more general result).

The lemniscate constant is given by the rapidly converging series

The constant is also given by the infinite product Also[41] A (generalized) continued fraction for π is

The lemniscate constant ϖ is related to the area under the curve

, twice the area in the positive quadrant under the curve

The lemniscate constant appears in the evaluation of the integrals

John Todd's lemniscate constants are defined by integrals:[9]

Euler discovered in 1738 that for the rectangular elastica (first and second lemniscate constants)[50][49]

This is also the arc length of the sine curve on half a period:[52]