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

1,2

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

References

B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.

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

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)

Example

1.3110287771460599052324197949455597068413774757158115814084108519...

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 *)

Prog

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

Crossrefs

Cf. A076390, A225119, A243340.

Sequence in context: A143119 A220419 A343006 * A216677 A341412 A196057

Adjacent sequences: A085562 A085563 A085564 * A085566 A085567 A085568

Keyword

nonn,cons

Author

N. J. A. Sloane, Jul 06 2003

Status

approved