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

1,2

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 and 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.

Amrik Singh Nimbran, Interesting infinite products of rational functions motivated by Euler, Math. Student, Vol. 85 (2016), pp. 117-133; ResearchGate link.

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).

Index entries for transcendental numbers.

Formula

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

From Jean-François Alcover, Apr 30 2013, updated Aug 01 2014: (Start)

Equals ellipticK(1/sqrt(2)).

Equals Pi/2*hypergeom([1/2,1/2],[1],1/2).

Equals the smallest positive root of cs(x/sqrt(2)|-1), where cs is the Jacobi elliptic function.

Equals the real half-period of the Weierstrass Pe function (Cf. Finch, p. 422). (End)

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

Equals Product_{k>=1} (4*k-2)*4*k/((4*k-3)*(4*k+1)) (Nimbran, 2016, p. 132, eq. (73)). - Amiram Eldar, Jun 28 2026

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 *)
(* Alternative: *)
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
(PARI) ellK(1/sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2025

Crossrefs

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

Sequence in context: A394396 A019609 A334502 * A134973 A030437 A200290

Adjacent sequences: A093338 A093339 A093340 * A093342 A093343 A093344

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