Lemniscate constant – Wikipedia

Ratio of the perimeter of Bernoulli’s lemniscate to its diameter

In mathematics, the lemniscate constant ϖ^[1][3][4][5] 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

${displaystyle (x^{2}+y^{2})^{2}=x^{2}-y^{2}}$

is 2ϖ. The lemniscate constant is closely related to the lemniscate elliptic functions and approximately equal to 2.62205755.^[6][7][8][9] The symbol ϖ is a cursive variant of π; see Pi § Variant pi.

Gauss’s constant, denoted by G, is equal to ϖ /π ≈ 0.8346268.^[10]

John Todd named two more lemniscate constants, the first lemniscate constant A = ϖ/2 ≈ 1.3110287771 and the second lemniscate constant B = π/(2ϖ) ≈ 0.5990701173.^{[11][12][13][14]}

Sometimes the quantities 2ϖ or A are referred to as the lemniscate constant.^[15][16]

History[edit]

Gauss’s constant

${displaystyle G}$

is named after Carl Friedrich Gauss, who calculated it via the arithmetic–geometric mean as

${displaystyle 1/M(1,{sqrt {2}})}$

.^[6] By 1799, Gauss had two proofs of the theorem that

${displaystyle M(1,{sqrt {2}})=pi /varpi }$

where

${displaystyle varpi }$

is the lemniscate constant.^[a]

The lemniscate constant

${displaystyle varpi }$

and first lemniscate constant

${displaystyle A}$

were proven transcendental by Theodor Schneider in 1937 and the second lemniscate constant

${displaystyle B}$

and Gauss’s constant

${displaystyle G}$

were proven transcendental by Theodor Schneider in 1941.^[11][17][b] In 1975, Gregory Chudnovsky proved that the set

${displaystyle {pi ,varpi }}$

is algebraically independent over

${displaystyle mathbb {Q} }$

, which implies that

${displaystyle A}$

and

${displaystyle B}$

are algebraically independent as well.^[18][19] But the set

${displaystyle {pi ,M(1,1/{sqrt {2}}),M'(1,1/{sqrt {2}})}}$

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

${displaystyle mathbb {Q} }$

. In fact,^[20]

Usually,

${displaystyle varpi }$

is defined by the first equality below.^[21][22]

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

${displaystyle M}$

,

Moreover,

which is analogous to

where

${displaystyle beta }$

is the Dirichlet beta function and

${displaystyle zeta }$

is the Riemann zeta function.^[23]

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

${displaystyle M(1,{sqrt {2}})}$

published in 1800:

Gauss’s constant is equal to

where Β denotes the beta function. A formula for G in terms of Jacobi theta functions is given by

Gauss’s constant may be computed from the gamma function at argument 1/4:

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

Viète’s formula for π can be written:

An analogous formula for ϖ is:^[25]

The Wallis product for π is:

An analogous formula for ϖ is:^[26]

A related result for Gauss’s constant (

${displaystyle G=varpi /pi }$

) is:^[27]

An infinite series of Gauss’s constant discovered by Gauss is:^[28]

The Machin formula for π is

${textstyle {tfrac {1}{4}}pi =4arctan {tfrac {1}{5}}-arctan {tfrac {1}{239}},}$

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

${textstyle {tfrac {1}{4}}pi =arctan {tfrac {1}{2}}+arctan {tfrac {1}{3}}}$

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

${displaystyle {tfrac {1}{2}}varpi =2operatorname {arcsl} {tfrac {1}{2}}+operatorname {arcsl} {tfrac {7}{23}}}$

, where

${displaystyle operatorname {arcsl} }$

is the lemniscate arcsine.^[29]

The lemniscate constant can be rapidly computed by the series^[30][31]

${displaystyle varpi =2^{-1/2}pi left(sum _{nin mathbb {Z} }e^{-pi n^{2}}right)^{2}=2^{1/4}pi e^{-pi /12}left(sum _{nin mathbb {Z} }(-1)^{n}e^{-pi p_{n}}right)^{2}}$

where

${displaystyle p_{n}=(3n^{2}-n)/2}$

(these are the generalized pentagonal numbers).

In a spirit similar to that of the Basel problem,

${displaystyle sum _{zin mathbb {Z} [i]setminus {0}}{frac {1}{z^{4}}}=G_{4}(i)={frac {varpi ^{4}}{15}}}$

where

${displaystyle mathbb {Z} [i]}$

are the Gaussian integers and

${displaystyle G_{4}}$

is the Eisenstein series of weight

${displaystyle 4}$

(see Lemniscate elliptic functions § Hurwitz numbers for a more general result).^[32]

A related result is

${displaystyle sum _{n=1}^{infty }sigma _{3}(n)e^{-2pi n}={frac {varpi ^{4}}{80pi ^{4}}}-{frac {1}{240}}}$

where

${displaystyle sigma _{3}}$

is the sum of positive divisors function.^[33]

In 1842, Malmsten found

${displaystyle sum _{n=1}^{infty }(-1)^{n+1}{frac {log(2n+1)}{2n+1}}={frac {pi }{4}}left(gamma +2log {frac {pi }{varpi {sqrt {2}}}}right)}$

where

${displaystyle gamma }$

is Euler’s constant.

Gauss’s constant is given by the rapidly converging series

The constant is also given by the infinite product

${displaystyle G=prod _{m=1}^{infty }tanh ^{2}left({frac {pi m}{2}}right).}$

Continued fractions[edit]

A (generalized) continued fraction for π is

An analogous formula for ϖ is^[12]

Define Brouncker’s continued fraction by^[34]