A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.

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

Offset: 1

Author

Jason Earls, Jun 25 2001

Keywords

• nonn
• cons
• easy

Examples

			2.622057554292119810464839589891119413682754951431623162816821703...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.3 and 6.2, pp. 99, 420.

Links

• Harry J. Smith, Table of n, a(n) for n = 1..5000
• Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits).
• Simon Plouffe, Lemniscate or Gauss constant.
• Simon Plouffe, Lemniscate constant or Gauss constant.
• Eric Weisstein's World of Mathematics, Lemniscate Constant.
• Wikipedia, Lemniscate constant.
• Index entries for transcendental numbers.

Crossrefs

Cf. A062540, A064853, A085565.

Cf. A002193, A014549, A093341.

Equals A000796/A053004 (see PARI script).

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // G. C. Greubel, Oct 07 2018

Maple

evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2,120); # Muniru A Asiru, Oct 08 2018
evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4),120); # Martin Renner, Aug 16 2019
evalf(1/2*Beta(1/4,1/2),120); # Martin Renner, Aug 16 2019
evalf(2*int(1/sqrt(1-x^4),x=0..1),120); # Martin Renner, Aug 16 2019

Mathematica

RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* Robert G. Wilson v, May 19 2004 *)

PARI

print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))

PARI

allocatemem(932245000); default(realprecision, 5080); x=Pi^(3/2)*sqrt(2)/(2*gamma(3/4)^2); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b062539.txt", n, " ", d)); \\ Harry J. Smith, Jun 20 2009

PARI

Pi/agm(1,sqrt(2)) \\ Charles R Greathouse IV, Feb 04 2015

PARI

intnum(x=0,Pi, 1/sqrt(1 + sin(x)^2)) \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.

A093341 multiplied by A002193. - R. J. Mathar, Aug 28 2013

From Martin Renner, Aug 16 2019: (Start)

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

Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)

Equals Pi/AGM(1, sqrt(2)). - Jean-François Alcover, Feb 28 2021

Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - Peter Bala, Mar 02 2022

Equals (1/2)*A064853 = 2*A085565. - Amiram Eldar, May 04 2022

Equals Pi*A014549. - Hugo Pfoertner, Jun 28 2024

Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - Stefano Spezia, Dec 15 2024

Equals Gamma(1/4)^2 / (sqrt(Pi)*2^(3/2)). - Vaclav Kotesovec, Apr 26 2025

Equals (161*6440^(1/4))/(2*Sum_{k>=0} N(k)/D(k)) with N(k) = Pochhammer(1/8,k) * Pochhammer(5/8,k) * (275+8640*k) and D(k) = (k!)^2*25921^k [Jorge Zuniga, 2023].