A130786 - OEIS

%I #20 Sep 21 2024 22:55:45

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

%T 7,1,5,9,9,9,5,5,7,0,2,6,0,5,5,4,0,1,0,3,7,2,6,5,2,9,2,1,3,7,1,4,9,5,

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

%N Decimal expansion of the complete elliptic integral of the first kind at sqrt(2)-1.

%H G. C. Greubel, <a href="/A130786/b130786.txt">Table of n, a(n) for n = 1..10000</a>

%H H. S. Wrigge, <a href="https://doi.org/10.1090/S0025-5718-1973-0324083-4">An Elliptic Integral Identity</a>, Math. Comp. 27 (1973) no 124, p <a href="http://www.ams.org/mathscinet-getitem?mr=0324083">839</a>.

%H I. J. Zucker and G. S. Joyce, <a href="https://doi.org/10.1017/S0305004101005254">Special values of the hypergeometric series II</a>, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319 (2.4)

%e Equals 1.64556839529345803986605168528707271599955702605540103726529213714...

%e which equals K[sqrt(2)-1] = Pi^(3/2)*sqrt[2+sqrt(2)]/(4*Gamma(5/8)*Gamma(7/8))

%e = 5.5683279... * 1.8477590650.. / ( 4 * 1.43451884..... * 1.0896523574...).

%p evalf(EllipticK(sqrt(2)-1));

%t RealDigits[Pi^(3/2)*Sqrt[2 + Sqrt@2]/(4 Gamma[5/8] Gamma[7/8]), 10, 111][[1]] (* _Robert G. Wilson v_, Jul 19 2007 *)

%t K[x_] := EllipticK[x^2/(x^2-1)]/Sqrt[1-x^2]; RealDigits[K[Sqrt[2]-1], 10, 111][[1]] (* _Jean-François Alcover_, Sep 22 2015 *)

%o (PARI) default(realprecision, 100); Pi^(3/2)*sqrt(2 + sqrt(2))/(4* gamma(5/8)*gamma(7/8)) \\ _G. C. Greubel_, Sep 27 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^(3/2)*Sqrt(2 + Sqrt(2))/(4*Gamma(5/8)*Gamma(7/8)); // _G. C. Greubel_, Sep 27 2018

%K cons,nonn

%O 1,2

%A _R. J. Mathar_, Jul 15 2007

%E More terms from _Robert G. Wilson v_, Jul 19 2007