Lemniscate constant

In mathematics, the lemniscate constant 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. The lemniscate constant is closely related to the lemniscate elliptic functions and is approximately equal to 2.62205755. It also appears in evaluation of the gamma and beta function at certain rational values. The symbol is a cursive variant of known as variant pi represented in Unicode by the character.
Sometimes the quantities or are referred to as the lemniscate constant.

History

Gauss's constant, denoted by G, is equal to and 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.
John Todd named two more lemniscate constants, the first lemniscate constant and the second lemniscate constant.
The lemniscate constant and Todd's first lemniscate constant were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd's 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 is not algebraically independent over. In 1996, Yuri Nesterenko proved that the set is algebraically independent over.
As of 2025 over 2 trillion digits of this constant have been calculated using y-cruncher.

Forms

Usually, is defined by the first equality below, but it has many equivalent forms:
where is the complete elliptic integral of the first kind with modulus, is the beta function, is the gamma function and is the Riemann zeta function.
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 published in 1800:John Todd's lemniscate constants may be given in terms of the beta function B:

As a special value of L-functions

which is analogous to
where is the Dirichlet beta function and is the Riemann zeta function.
Analogously to the Leibniz formula for π,
we have
where is the L-function of the elliptic curve over ; this means that is the multiplicative function given by
where is the number of solutions of the congruence
in variables that are non-negative integers.
Equivalently, is given by
where such that and is the eta function.
The above result can be equivalently written as
and also tells us that the BSD conjecture is true for the above.
The first few values of are given by the following table; if such that doesn't appear in the table, then :

As a special value of other functions

Let be the minimal weight level new form. Then
The -coefficient of is the Ramanujan tau function.

Series

Viète's formula for can be written:
An analogous formula for is:
The Wallis product for is:
An analogous formula for is:
A related result for Gauss's constant is:
An infinite series discovered by Gauss is:
The Machin formula for is and several similar formulas for can be developed using trigonometric angle sum identities, e.g. Euler's formula. Analogous formulas can be developed for, including the following found by Gauss:, where is the lemniscate arcsine.
The lemniscate constant can be rapidly computed by the series
where . Also
In a spirit similar to that of the Basel problem,
where are the Gaussian integers and is the Eisenstein series of weight .
A related result is
where is the sum of positive divisors function.
In 1842, Malmsten found
where is Euler's constant and is the Dirichlet-Beta function.
The lemniscate constant is given by the rapidly converging series
The constant is also given by the infinite product
Also

Continued fractions

A continued fraction for is
An analogous formula for is
Define Brouncker's continued fraction by
Let except for the first equality where. Then
For example,
In fact, the values of and, coupled with the functional equation
determine the values of for all.

Simple continued fractions

Simple continued fractions for the lemniscate constant and related constants include

Integrals

The lemniscate constant is related to the area under the curve. Defining, twice the area in the positive quadrant under the curve is In the quartic case,
In 1842, Malmsten discovered that
Furthermore,
and
a form of Gaussian integral.
The lemniscate constant appears in the evaluation of the integrals
John Todd's lemniscate constants are defined by integrals:

Circumference of an ellipse

The lemniscate constant satisfies the equation
Euler discovered in 1738 that for the rectangular elastica
Now considering the circumference of the ellipse with axes and, satisfying, Stirling noted that
Hence the full circumference is
This is also the arc length of the sine curve on half a period:

Other limits

Analogously to
where are Bernoulli numbers, we have
where are Hurwitz numbers.