A093341 Decimal expansion of "lemniscate case".
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, 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, 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
Offset: 1
Examples
1.854074677301371918433850347195260046217598823521766905585928045056021...
References
- 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.
- Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software, 1990, p. iii.
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 6.1 Gauss' Lemniscate Constant, pp. 421-422.
Links
- Harry J. Smith, Table of n, a(n) for n = 1..5000
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Section 18.14.7, p. 658.
- G. Mingari Scarpello and D. Ritelli, On computing some special values of hypergeometric functions, arXiv:1212.0251 [math.CA], 2012-2014, eq. (4.1).
- Eric Weisstein's World of Mathematics, Lemniscate Case.
- I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319, (2.3).
Programs
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Maple
evalf( EllipticK(1/sqrt(2)) ); # R. J. Mathar, Aug 28 2013
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Mathematica
RealDigits[ N[ Gamma[1/4]^2 / (4*Sqrt[Pi]), 105]][[1]] (* Jean-François Alcover, Oct 04 2011 *) RealDigits[N[EllipticK[1/2], 105]][[1]] (* Vaclav Kotesovec, Feb 22 2015 *)
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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 -
PARI
Pi/agm(sqrt(2),2) \\ Charles R Greathouse IV, Feb 04 2015
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PARI
ellK(1/sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2025
Formula
GAMMA(1/4)^2/(4*(Pi)^(1/2)). - Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
Also equals ellipticK(1/sqrt(2)) = Pi/2*hypergeom([1/2,1/2],[1],1/2),
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
From Peter Bala, Feb 22 2015: (Start)
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.
Equals 2 * Sum {n >= 0} (-1/4)^n * binomial(2*n,n) * 1/(4*n + 1). (End)
Equals A062539 / sqrt(2). - Amiram Eldar, May 04 2022
Extensions
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004