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A085565 Decimal expansion of lemniscate constant A.

%I A085565 #97 Feb 16 2025 08:32:50
%S A085565 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,
%T A085565 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,
%U A085565 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
%N A085565 Decimal expansion of lemniscate constant A.
%C A085565 This number is transcendental by a result of Schneider on elliptic integrals. - _Benoit Cloitre_, Jan 08 2006
%C A085565 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
%C A085565 Also the ratio of generating curve length to diameter of a "Mylar balloon" (see Paulsen). - _Jeremy Tan_, May 05 2021
%D A085565 B. C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, p. 140, Entry 25.
%D A085565 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.1, p. 421.
%D A085565 Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).
%D A085565 Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).
%H A085565 G. C. Greubel, <a href="/A085565/b085565.txt">Table of n, a(n) for n = 1..5000</a>
%H A085565 John Maxwell Campbell, <a href="http://math.colgate.edu/~integers/v107/v107.pdf">WZ proofs for lemniscate-like constant evaluations</a>, Integers 21 (2021), Article A107, 15.
%H A085565 S. Khrushchev, <a href="http://www.maths.ed.ac.uk/~aar/papers/khrushchev.pdf">Orthogonal polynomials and continued fractions from Euler’s point of view</a>, Encyclopedia of Mathematics and its Applications 122.
%H A085565 Rensley Meulens, <a href="https://doi.org/10.1063/5.0074083">A note on N-soliton solutions for the viscid incompressible Navier-Stokes differential equation</a>, AIP Advances (2022) Vol. 12, 015308.
%H A085565 W. H. Paulsen, <a href="https://doi.org/10.2307/2975161">What Is the Shape of a Mylar Balloon?</a>, Amer. Math. Monthly 101 (10), (Dec. 1994), pp. 953-958.
%H A085565 J. Todd, <a href="https://doi.org/10.1145/360569.360580">The lemniscate constants</a>, Comm. ACM, 18 (1975), 14-19; 18 (1975), 462.
%H A085565 J. Todd, <a href="https://doi.org/10.1007/978-1-4757-3240-5_45">The lemniscate constants</a>, in Pi: A Source Book, pp. 412-417.
%H A085565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>.
%H A085565 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mylar_balloon_(geometry)">Mylar balloon</a>.
%H A085565 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A085565 Equals (1/4)*(2*Pi)^(-1/2)*GAMMA(1/4)^2.
%F A085565 Equals Integral_{x>=1}dx/sqrt(4x^3-4x). - _Benoit Cloitre_, Jan 08 2006
%F A085565 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]
%F A085565 Equals Pi/sqrt(8)/agm(1,sqrt(1/2)).
%F A085565 Equals Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).
%F A085565 Product_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - _Jean-François Alcover_, Sep 02 2014, after Steven Finch
%F A085565 From _Peter Bala_, Mar 09 2015: (Start)
%F A085565 Equals Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.
%F A085565 Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179.
%F A085565 Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End)
%F A085565 From _Peter Bala_, Oct 25 2019: (Start)
%F A085565 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.
%F A085565 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.
%F A085565 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)
%F A085565 Equals (3/2)*A225119. - _Peter Bala_, Oct 27 2019
%F A085565 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
%F A085565 From  _Peter Bala_, Mar 24 2024: (Start)
%F A085565 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.
%F A085565 For example, taking k = 0 and k = 1 yields
%F A085565 A = 2/(1 + (1*3)/(2 + (5*7)/(2 + (9*11)/(2 + (13*15)/(2 + ... + (4*n + 1)*(4*n + 3)/(2 + ... )))))) and
%F A085565 A = (1/4)*(5 + (1*3)/(10 + (5*7)/(10 + (9*11)/(10 + (13*15)/(10 + ... + (4*n + 1)*(4*n + 3)/(10 + ... )))))). (End)
%e A085565 1.3110287771460599052324197949455597068413774757158115814084108519...
%t A085565 RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]]
%t A085565 (* or *)
%t A085565 RealDigits[ EllipticK[-1], 10, 99][[1]] (* _Jean-François Alcover_, Mar 07 2013, updated Jul 30 2016 *)
%o A085565 (PARI) gamma(1/4)^2/4/sqrt(2*Pi)
%o A085565 (PARI) K(x)=Pi/2/agm(1,sqrt(1-x))
%o A085565 K(-1) \\ _Charles R Greathouse IV_, Aug 02 2018
%o A085565 (PARI) ellK(I) \\ _Charles R Greathouse IV_, Feb 04 2025
%o A085565 (Magma) C<i> := ComplexField(); [Gamma(1/4)^2/(4*Sqrt(2*Pi(C)))]; // _G. C. Greubel_, Nov 05 2017
%Y A085565 Cf. A076390, A225119, A243340.
%K A085565 nonn,cons
%O A085565 1,2
%A A085565 _N. J. A. Sloane_, Jul 06 2003