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A085565 Decimal expansion of lemniscate constant A. 8
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This number is transcendental by a result of Schneider on elliptic integrals. - Benoit Cloitre, Jan 08 2006

REFERENCES

Th. Schneider, Transzendenzuntersuchungen periodischer Funktionen (1934).

Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale (1937).

J. Todd, The lemniscate constants, Comm. ACM, 18 (1975), 14-19; 18 (1975), 462.

LINKS

Table of n, a(n) for n=1..99.

S. Khrushchev, Orthogonal polynomials and continued fractions from Euler’s point of view, Encyclopedia of Mathematics and its Applications 122

Eric Weisstein's World of Mathematics, Lemniscate Constant

FORMULA

(1/4)*(2*Pi)^(-1/2)*GAMMA(1/4)^2.

Integral_{1}^{infty}dx/sqrt(4x^3-4x)=Gamma(1/4)^2/4/sqrt(2*Pi)= 1.31102877714605990523... . - Benoit Cloitre, Jan 08 2006

Equals prod(k>=0, [(4k+3)(4k+2)] / [(4k+5)(4k+4)] ) (Gauss). - Ralf Stephan, Mar 04 2008

Pi/sqrt(8)/agm(1,sqrt(1/2)).

Pi/sqrt(8)*hypergeom([1/2,1/2],[1],1/2).

Prod_{m>=1} ((2*m)/(2*m+1))^(-1)^m. - Jean-François Alcover, Sep 02 2014, after Steven Finch.

From Peter Bala, Mar 09 2015: (Start)

Equals int_{x = 0..1} 1/sqrt(1 - x^4) dx.

Continued fraction representations: 2/(1 + 1*3/(2 + 5*7/(2 + 9*11/(2 + ... )))) due to Euler - see Khrushchev, p. 179.

Also equals 1 + 1/(2 + 2*3/(2 + 4*5/(2 + 6*7/(2 + ... )))). (End)

EXAMPLE

1.3110287771460599052324197949455597068413774757158115814084108519...

MATHEMATICA

RealDigits[ Gamma[1/4]^2/(4*Sqrt[2*Pi]), 10, 99][[1]]

(* or *)

RealDigits[ EllipticK[-1], 10, 99][[1]] (* Jean-François Alcover, Mar 07 2013, updated Jul 30 2016 *)

PROG

(PARI) gamma(1/4)^2/4/sqrt(2*Pi)

CROSSREFS

Cf. A076390.

Sequence in context: A011354 A143119 A220419 * A216677 A196057 A058395

Adjacent sequences:  A085562 A085563 A085564 * A085566 A085567 A085568

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane, Jul 06 2003

STATUS

approved

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Last modified April 27 15:06 EDT 2017. Contains 285528 sequences.