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
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
KEYWORD
nonn,cons
AUTHOR
N. J. A. Sloane, Jul 06 2003
STATUS
approved