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Lemniscate Constant

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The lemniscate constant is the quantity L=1/2s, where

s=1/(sqrt(2pi))[Gamma(1/4)]^2=5.2441151085...
(1)

(OEIS A064853) is the arc length of a lemniscate with a=1. It can be written

L=1/2s
(2)
=int_0^pi(dtheta)/(sqrt(1+sin^2theta))
(3)
=2int_0^1(dx)/(sqrt(1-x^4))
(4)
=piG
(5)
=pi/(M(1,sqrt(2)))
(6)
=2K(i)
(7)
=sqrt(2)K(1/(sqrt(2)))
(8)
=([Gamma(1/4)]^2)/(2sqrt(2pi))
(9)
=pitheta_4^2(e^(-pi))
(10)
=(3pi)/(2R_D(0,2,1))
(11)
=2R_F(0,1,2)
(12)
=piR_K(1,2)
(13)
=2.62205755429...
(14)

(OEIS A062539; Abramowitz and Stegun 1972; Finch 2003, p. 420), where G is Gauss's constant, M(a,b) is the arithmetic-geometric mean, theta_4(q) is a Jacobi theta function, K(k) is a complete elliptic integral of the first kind, and R_D, R_F, and R_K are Carlson elliptic integrals. Todd (1975) cites T. Schneider as proving L to be a transcendental number in 1937.

The quantity

L_1=1/2L
(15)
=int_0^1(dx)/(sqrt(1-x^4))
(16)
=1.311028777...
(17)

(OEIS A085565; Le Lionnais 1983) is sometimes known as the first lemniscate constant, while

L_2=int_0^1(x^2)/(sqrt(1-x^4))dx
(18)
=pi/(2L)
(19)
=1/(2G)
(20)
=0.5990701173...
(21)

(OEIS A076390), where G is Gauss's constant, is sometimes known as the second lemniscate constant (Todd 1975, Gosper 1976, Lewanowicz and Paszowski 1995).

A convergent series given by Zuniga (2023, p. 10), in the normalization used here, is

L=1/2sum_(n=1)^infty((-1)/(27))^n((224n-144)(1/4)_n(3/4)_n)/(-(4n-1)(4n-3)(1/12)_n(5/12)_n),
(22)

where (a)_n is the Pochhammer symbol.

Chudnovsky (2026) announced that the irrationality measure of L is 2. The paper uses the symbol Omega for the same constant and reports mathematical collaboration with ChatGPT and a Lean formalization. The complete proof and a clean-cache build of the reported formalization have not been independently verified; the assertion is therefore an announced result, not an established value.

Chudnovsky and Chudnovsky (2026) announced the broader assertion that every nonzero real period of an elliptic curve over Q with complex multiplication, measured using a specified nonzero Q-rational invariant differential, has irrationality measure 2. For powers of the lemniscate constant they separately report mu(L^n)=O(nln(2n)) for integers n>=1, where mu denotes irrationality measure; this is not a claim that every power has measure 2. Their reported Lean verification retains explicit classical uniformization, complex-multiplication, and theta-function inputs and has not been independently rebuilt. Chudnovsky and Chudnovsky (2026) report ChatGPT assistance with mathematical exploration and formalization.

See also

Gamma Function, Lemniscate, Lemniscate Case, Pseudolemniscate Case

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Chudnovsky, G. V. "The Irrationality Exponent of the Lemniscate Period: A Determinant Proof with a Lean 4 Formalization." 19 Oct 2026. https://arxiv.org/abs/2610.26539.Chudnovsky, G. V. and Chudnovsky, D. V. "Irrationality Exponents of Real CM Elliptic Periods and Powers of the Lemniscate Period." 22 Oct 2026. https://arxiv.org/abs/2610.31112.Finch, S. R. "Gauss' Lemniscate Constant." §6.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 420-423, 2003.Gosper, R. W. "A Calculus of Series Rearrangements." In Algorithms and Complexity: New Directions and Recent Results. Proc. 1976 Carnegie-Mellon Conference (Ed. J. F. Traub). New York: Academic Press, pp. 121-151, 1976.Le Lionnais, F. Les nombres remarquables. Paris, France: Hermann, p. 37, 1983.Levin, A. "A Geometric Interpretation of an Infinite Product for the Lemniscate Constants." Amer. Math. Monthly 113, 510-520, 2006.Lewanowicz, S. and Paszowski, S. "An Analytic Method for Convergence Acceleration of Certain Hypergeometric Series." Math. Comput. 64, 691-713, 1995.Sloane, N. J. A. Sequences A062539, A064853, A076390, and A085565 in "The On-Line Encyclopedia of Integer Sequences."Todd, J. "The Lemniscate Constant." Comm. ACM 18, 14-19 and 462, 1975.Zuniga, J. "On Fast Computing Formulas for Apéry's and Lemniscate Constants." Unpublished report, Oct. 2023.

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Lemniscate Constant

Cite this as:

Weisstein, Eric W. "Lemniscate Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LemniscateConstant.html

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