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 for the circle.See: p. 404
p. 199 p. 57 p. 203 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.See:
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.Neither of these proofs was rigorous from the modern point of view. See
John Todd named two more lemniscate constants, the first lemniscate constant and the second lemniscate constant . and
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 particular, Siegel proved that if and with are algebraic, then or is transcendental. Here, and are Eisenstein series. The fact that is transcendental follows from and In particular, Schneider proved that the beta function is transcendental for all such that . The fact that is transcendental follows from and similarly for and from In 1975, Gregory Chudnovsky proved that and are algebraically independent over , which implies that and are algebraically independent as well.G. V. Choodnovsky: Algebraic independence of constants connected with the functions of analysis, Notices of the AMS 22, 1975, p. A-486G. V. Chudnovsky: Contributions to The Theory of Transcendental Numbers, American Mathematical Society, 1984, p. 6 But the set (where the prime denotes the derivative with respect to the second variable) is not algebraically independent over . p. 45 In fact,
In 1996, Yuri Nesterenko proved that , , and are 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:See:
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 p. 31, formula (2.8.10)In fact, the series converges for .
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 ( is the set of all primes). Equivalently, is given by
where such that and is the eta function.The function is the unique weight level new form and it satisfies the functional equationThe function is closely related to the function which is the multiplicative function defined by
where is the number of solutions of the equation
in variables that are non-negative integers (see Fermat's theorem on sums of two squares) and is the Dirichlet character from the Leibniz formula for π; also
for any positive integer where the sum extends only over positive divisors; the relation between and is
where is any non-negative integer.The function also appears in
where is any positive integer and is the set of all Gaussian integers of the form
where is odd and is even. The function from the previous note satisfies
where is positive odd. The above result can be equivalently written as
(the number is the conductor of ) 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:Levin (2006)
The Wallis product for is:
An analogous formula for is:Hyde (2014) proves the validity of a more general Wallis-like formula for clover curves; here the special case of the lemniscate is slightly transformed, for clarity.
A related result for Gauss's constant () is:
An infinite series discovered by Gauss is: p. 60
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.Todd (1975)
The lemniscate constant can be rapidly computed by the series for the first equality. The second equality can be proved by using the pentagonal number theorem. p. 326
where (these are the generalized pentagonal numbers). AlsoThis formula can be proved by hypergeometric inversion: Let
where with . Then
where
where . The formula in question follows from setting .
In a spirit similar to that of the Basel problem,
where are the Gaussian integers and is the Eisenstein series of weight (see Lemniscate elliptic functions § Hurwitz numbers for a more general result). p. 232
A related result is
where is the sum of positive divisors function. p. 11—13
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
AlsoThe formula follows from the hypergeometric transformation
where and is the modular lambda function.
Continued fractions A (generalized) continued fraction for is
An analogous formula for is
Define Brouncker's continued fraction by p. 140 (eq. 3.34), p. 153. There's an error on p. 153: should be .
Let except for the first equality where . Then p. 146, 155 p. 36, eq. 24
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 (first and second lemniscate constants)Levien (2008)
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: In this paper and .
Other limits
Analogously to
where are Bernoulli numbers, we have
where are Hurwitz numbers.
Notes
References
External links
Sequences A014549, A053002, and A062539 in OEIS