cp's OEIS Frontend

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

%I A062539 #87 Jul 04 2025 14:58:22
%S A062539 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,
%T A062539 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,
%U A062539 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
%N A062539 Decimal expansion of the Lemniscate constant or Gauss's constant.
%D A062539 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.
%H A062539 Harry J. Smith, <a href="/A062539/b062539.txt">Table of n, a(n) for n = 1..5000</a>
%H A062539 Lorenz Milla, <a href="https://archive.org/details/lemniscate_digits">World record computation of Lemniscate Constant (2,000,000,000,000 digits)</a>.
%H A062539 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/lemni.txt">Lemniscate or Gauss constant</a>.
%H A062539 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap53.html">Lemniscate constant or Gauss constant</a>.
%H A062539 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>.
%H A062539 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lemniscate_elliptic_functions#Lemniscate_constant">Lemniscate constant</a>.
%H A062539 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A062539 Equals (1/2)*sqrt(2*Pi^3)/Gamma(3/4)^2.
%F A062539 A093341 multiplied by A002193. - _R. J. Mathar_, Aug 28 2013
%F A062539 From _Martin Renner_, Aug 16 2019: (Start)
%F A062539 Equals 2*Integral_{x=0..1} 1/sqrt(1-x^4) dx.
%F A062539 Equals 1/2*B(1/4,1/2) with Beta function B(x,y) = Gamma(x)*Gamma(y)/Gamma(x+y). (End)
%F A062539 Equals Pi/AGM(1, sqrt(2)). - _Jean-François Alcover_, Feb 28 2021
%F A062539 Equals 2*hypergeom([1/2, 1/4], [5/4], 1). - _Peter Bala_, Mar 02 2022
%F A062539 Equals (1/2)*A064853 = 2*A085565. - _Amiram Eldar_, May 04 2022
%F A062539 Equals Pi*A014549. - _Hugo Pfoertner_, Jun 28 2024
%F A062539 Equals Integral_{x=0..Pi} 1/sqrt(1 + sin(x)^2) dx = EllipticK(-1) (see Finch at p. 420). - _Stefano Spezia_, Dec 15 2024
%F A062539 Equals Gamma(1/4)^2 / (sqrt(Pi)*2^(3/2)). - _Vaclav Kotesovec_, Apr 26 2025
%F A062539 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].
%e A062539 2.622057554292119810464839589891119413682754951431623162816821703...
%p A062539 evalf((1/2)*sqrt(2*Pi^3)/GAMMA(3/4)^2,120); # _Muniru A Asiru_, Oct 08 2018
%p A062539 evalf(1/2*GAMMA(1/4)*GAMMA(1/2)/GAMMA(3/4),120); # _Martin Renner_, Aug 16 2019
%p A062539 evalf(1/2*Beta(1/4,1/2),120); # _Martin Renner_, Aug 16 2019
%p A062539 evalf(2*int(1/sqrt(1-x^4),x=0..1),120); # _Martin Renner_, Aug 16 2019
%t A062539 RealDigits[Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2, 10, 111][[1]] (* _Robert G. Wilson v_, May 19 2004 *)
%o A062539 (PARI) print(1/2*Pi^(3/2)/gamma(3/4)^2*2^(1/2))
%o A062539 (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
%o A062539 (PARI) Pi/agm(1,sqrt(2)) \\ _Charles R Greathouse IV_, Feb 04 2015
%o A062539 (PARI) intnum(x=0,Pi, 1/sqrt(1 + sin(x)^2)) \\ _Charles R Greathouse IV_, Feb 04 2025
%o A062539 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (1/2)*Sqrt(2*Pi(R)^3)/Gamma(3/4)^2; // _G. C. Greubel_, Oct 07 2018
%Y A062539 Cf. A062540, A064853, A085565.
%Y A062539 Cf. A002193, A014549, A093341.
%Y A062539 Equals A000796/A053004 (see PARI script).
%K A062539 nonn,cons,easy
%O A062539 1,1
%A A062539 _Jason Earls_, Jun 25 2001