A085565 Decimal expansion of lemniscate constant A.

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

Offset: 1

Author

N. J. A. Sloane, Jul 06 2003

Keywords

• nonn
• cons

Comments

This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006

The two lemniscate constants are A = Integral_{x = 0..1} 1/sqrt(1 - x^4) dx and B = Integral_{x = 0..1} x^2/sqrt(1 - x^4) dx. See A076390. - Peter Bala, Oct 25 2019

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

Examples

			1.3110287771460599052324197949455597068413774757158115814084108519...
		

References

• B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
• Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).
• Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• John Maxwell Campbell, WZ proofs for lemniscate-like constant evaluations, Integers 21 (2021), Article A107, 15.
• S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122.
• Rensley Meulens, A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation, AIP Advances (2022) Vol. 12, 015308.
• 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, 18 (1975), 14-19; 18 (1975), 462.
• J. Todd, The lemniscate constants, in Pi: A Source Book, pp. 412-417.
• Eric Weisstein's World of Mathematics, Lemniscate Constant.
• Wikipedia, Mylar balloon.
• Index entries for transcendental numbers.

Crossrefs

Cf. A076390, A225119, A243340.

Programs

Magma

C := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // G. C. Greubel, Nov 05 2017

Mathematica

RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]]
(* or *)
RealDigits[ EllipticK[-1], 10, 99][[1]] (* Jean-François Alcover, Mar 07 2013, updated Jul 30 2016 *)

PARI

gamma(1/4)^2/4/sqrt(2*Pi)

PARI

K(x)=Pi/2/agm(1,sqrt(1-x))
K(-1) \\ Charles R Greathouse IV, Aug 02 2018

PARI

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

Formula

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

Equals Integral_{x>=1}dx/sqrt(4x^3-4x). - Benoit Cloitre, Jan 08 2006

Equals Product_(k>=0, [(4k+3)(4k+4)] / [(4k+5)(4k+2)] ) (Gauss). - Ralf Stephan, Mar 04 2008 [corrected by Vaclav Kotesovec, May 01 2020]

Equals Pi/sqrt(8)/agm(1,sqrt(1/2)).

Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).

Product_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch

From Peter Bala, Mar 09 2015: (Start)

Equals Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.

Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179.

Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End)

From Peter Bala, Oct 25 2019: (Start)

Equals 1 + 1/5 + (1*3)/(5*9) + (1*3*5)/(5*9*13) + ... = hypergeom([1/2,1],[5/4],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 - 3)*r(n-1) with r(1) = 1. Then the constant equals Sum_{n >= 1} r(n) = 1 + 1/5 + 1/15 + 1/39 + 7/663 + 1/221 + 11/5525 + 11/12325 + 1/2465 + .... The partial sum of the series to 100 terms gives 32 correct decimal digits for the constant.

Equals (1*3)/(1*5) + (1*3*5)/(1*5*9) + (1*3*5*7)/(1*5*9*13) + ... = (3/5) * hypergeom([5/2,1],[9/4],1/2). (End)

Equals (3/2)*A225119. - Peter Bala, Oct 27 2019

Equals Integral_{x=0..Pi/2} 1/sqrt(1 + cos(x)^2) dx = Integral_{x=0..Pi/2} 1/sqrt(1 + sin(x)^2) dx. - Amiram Eldar, Aug 09 2020

From Peter Bala, Mar 24 2024: (Start)

An infinite family of continued fraction expansions for this constant can be obtained from Berndt, Entry 25, by setting n = 1/2 and x = 4*k + 1 for k >= 0.

For example, taking k = 0 and k = 1 yields

A = 2/(1 + (1*3)/(2 + (5*7)/(2 + (9*11)/(2 + (13*15)/(2 + ... + (4*n + 1)*(4*n + 3)/(2 + ... )))))) and

A = (1/4)*(5 + (1*3)/(10 + (5*7)/(10 + (9*11)/(10 + (13*15)/(10 + ... + (4*n + 1)*(4*n + 3)/(10 + ... )))))). (End)