%I #67 Aug 27 2024 02:46:37

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

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

%U 2,1,7,7,6,8,3,8,1,1,9,9,7,8,3,5,7,2,7,1,8,6,1,6,5,0,3,7,1,8,9,7,2,7,7,7,7

%N Decimal expansion of "lemniscate case".

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th printing. New York: Dover, 1972, Section 18.14.7, p. 658.

%D Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software, 1990, p. iii.

%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.1 Gauss' Lemniscate Constant, p. 421.

%H Harry J. Smith, <a href="/A093341/b093341.txt">Table of n, a(n) for n = 1..5000</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=658">Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables</a>, Section 18.14.7, p. 658.

%H G. Mingari Scarpello and D. Ritelli, <a href="http://arxiv.org/abs/1212.0251">On computing some special values of hypergeometric functions</a>, arXiv:1212.0251 [math.CA], 2012-2014, eq. (4.1).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateCase.html">Lemniscate Case</a>.

%H I. J. Zucker and G. S. Joyce, <a href="https://doi.org/10.1017/S0305004101005254">Special values of the hypergeometric series II</a>, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319, (2.3).

%F GAMMA(1/4)^2/(4*(Pi)^(1/2)). - Pab Ter (pabrlos(AT)yahoo.com), May 24 2004

%F Also equals ellipticK(1/sqrt(2)) = Pi/2*hypergeom([1/2,1/2],[1],1/2),

%F or also the smallest positive root of cs(x/sqrt(2)|-1), where cs is the Jacobi elliptic function, or also the real half-period of the Weierstrass Pe function (Cf. Finch p. 422). - _Jean-François Alcover_, Apr 30 2013, updated Aug 01 2014

%F From _Peter Bala_, Feb 22 2015: (Start)

%F Equals Integral_{x = 0..oo} 1/sqrt(1 + x^4) dx = 2 * Integral_{x = 0..1} 1/sqrt(1 + x^4) dx = sqrt(2) * Integral_{x = 0..1} 1/sqrt(1 - x^4) dx.

%F Equals 2 * Sum {n >= 0} (-1/4)^n * binomial(2*n,n) * 1/(4*n + 1). (End)

%F Equals A062539 / sqrt(2). - _Amiram Eldar_, May 04 2022

%F Equals 1/A105372 = A175576/2 = 2*A224268. - _Hugo Pfoertner_, Aug 27 2024

%e 1.854074677301371918433850347195260046217598823521766905585928045056021...

%p evalf( EllipticK(1/sqrt(2)) ); # _R. J. Mathar_, Aug 28 2013

%t RealDigits[ N[ Gamma[1/4]^2 / (4*Sqrt[Pi]), 105]][[1]] (* _Jean-François Alcover_, Oct 04 2011 *)

%t RealDigits[N[EllipticK[1/2], 105]][[1]] (* _Vaclav Kotesovec_, Feb 22 2015 *)

%o (PARI) { allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/(4*(Pi)^(1/2)); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b093341.txt", n, " ", d)); } \\ _Harry J. Smith_, Jun 19 2009

%o (PARI) Pi/agm(sqrt(2),2) \\ _Charles R Greathouse IV_, Feb 04 2015

%Y Cf. A064853, A062539, A105372, A153396, A175576, A224268.

%K cons,nonn,easy

%O 1,2

%A Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 26 2004

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004