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A093341
Decimal expansion of "lemniscate case".
16
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
1,2
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, p. 421.
LINKS
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).
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
Equals 1/A105372 = A175576/2 = 2*A224268. - Hugo Pfoertner, Aug 27 2024
EXAMPLE
1.854074677301371918433850347195260046217598823521766905585928045056021...
MAPLE
evalf( EllipticK(1/sqrt(2)) ); # R. J. Mathar, Aug 28 2013
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 *)
PROG
(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
CROSSREFS
KEYWORD
cons,nonn,easy
AUTHOR
Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Apr 26 2004
EXTENSIONS
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 24 2004
STATUS
approved