A076390 Decimal expansion of lemniscate constant B.

5, 9, 9, 0, 7, 0, 1, 1, 7, 3, 6, 7, 7, 9, 6, 1, 0, 3, 7, 1, 9, 9, 6, 1, 2, 4, 6, 1, 4, 0, 1, 6, 1, 9, 3, 9, 1, 1, 3, 6, 0, 6, 3, 3, 1, 6, 0, 7, 8, 2, 5, 7, 7, 9, 1, 3, 1, 8, 3, 7, 4, 7, 6, 4, 7, 3, 2, 0, 2, 6, 0, 7, 0, 7, 1, 9, 5, 7, 8, 3, 5, 4, 1, 7, 9, 4, 2, 7, 7, 8, 2, 4, 4, 8, 9, 6, 6, 9, 4, 6, 8, 7, 9, 5, 3, 6

Offset: 0

Author

Robert G. Wilson v, Oct 09 2002

Keywords

• nonn
• cons

Comments

Also decimal expansion of AGM(1,i)/(1+i).

See A085565 for the lemniscate constant A. - Peter Bala, Oct 25 2019

Also the ratio of height to diameter of a "Mylar balloon" (see Paulsen). - Jeremy Tan, May 05 2021

Examples

			0.599070117367796103719961246140161939113606331607825779131837476473202607...
AGM(1,i) = 0.59907011736779610371... + 0.59907011736779610371...*i
		

References

• J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1998.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.

Links

• W. H. Paulsen, What Is the Shape of a Mylar Balloon?, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.
• J. Todd, The lemniscate constants, Comm. ACM, Vol. 18, No. 1 (1975), pp. 14-19; corrigendum, Vol. 18, No. 8 (1975), p. 462.
• J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
• Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean.
• Eric Weisstein's World of Mathematics, Lemniscate Constant.
• Wikipedia, Mylar balloon.
• Wolfram Research, Arithmetic-Geometric Mean.

Crossrefs

Cf. A053004, A076391, A076392, A085565, A243340.

Programs

Mathematica

RealDigits[ Chop[ N[ ArithmeticGeometricMean[1, I]/(1 + I), 111]]] [[1]]
RealDigits[N[Pi/(4 EllipticK[-1]), 106]][[1]] (* Jean-François Alcover, Jun 02 2019 *)

PARI

real(agm(1,I)/(1+I)) \\ Charles R Greathouse IV, Mar 03 2016

PARI

(2*Pi)^(-1/2)*gamma(3/4)^2 \\ Michel Marcus, Nov 10 2017

PARI

Pi/4/ellK(I) \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals (2*Pi)^(-1/2)*GAMMA(3/4)^2.

Equals ee/sqrt(2)-1/2*sqrt(2*ee^2-Pi) where ee = EllipticE(1/2), or also Product_{m>=1} ((2*m)/(2*m-1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch.

Equals sqrt(2) * Pi^(3/2) / GAMMA(1/4)^2. - Vaclav Kotesovec, Oct 03 2019

From Peter Bala, Oct 25 2019: (Start)

Equals 1 - 1/3 - 1/(3*7) - (1*3)/(3*7*11) - (1*3*5)/(3*7*11*15) - ... = hypergeom([-1/2,1],[3/4],1/2) by Gauss’s second summation theorem.

Equivalently, define a sequence of rational numbers r(n) recursively by r(n) = (2*n - 3)/(4*n - 1)*r(n-1) with r(0) = 1. Then the constant equals Sum_{n >= 0} r(n) = 1 - 1/3 - 1/21 - 1/77 - 1/231 - 1/627 - 3/4807 - 1/3933 - 13/121923 - 13/284487 - 17/853461 - .... The partial sum of the series to 100 terms gives the constant correct to 32 decimal places.

Equals (1/3) + (1*3)/(3*7) + (1*3*5)/(3*7*11) + ... = (1/3) * hypergeom ([3/2,1],[7/4],1/2). (End)

Equals (1/2) * A053004. - Amiram Eldar, Aug 26 2020

Equals (2/3) * 1/A243340. - Peter Bala, Mar 25 2024

Equals Product_{n>=1} exp(((-1)^n*beta(n))/n), where beta(n) is the Dirichlet beta function. - Antonio Graciá Llorente, Oct 16 2024

Equals Integral_{x=0..1} x^2/sqrt(1 - x^4) dx = sqrt(Pi)*Gamma(7/4)/(3*Gamma(5/4)) (see Finch). - Stefano Spezia, Dec 15 2024

Extensions

Edited by N. J. A. Sloane, Nov 01 2008 at the suggestion of R. J. Mathar