Lemniscate Constant



(OEIS A064853) be the arc length of a lemniscate with a=1. Then the lemniscate constant is the quantity


(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 theta_4 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


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


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

See also

Gamma Function, Lemniscate, Lemniscate Case, Pseudolemniscate Case

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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.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: 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.

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Lemniscate Constant

Cite this as:

Weisstein, Eric W. "Lemniscate Constant." From MathWorld--A Wolfram Web Resource.

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