A130786 Decimal expansion of the complete elliptic integral of the first kind at sqrt(2)-1.

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, 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, 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

Offset: 1

Author

R. J. Mathar, Jul 15 2007

Keywords

• cons
• nonn

Examples

			Equals 1.64556839529345803986605168528707271599955702605540103726529213714...
which equals K[sqrt(2)-1] = Pi^(3/2)*sqrt[2+sqrt(2)]/(4*Gamma(5/8)*Gamma(7/8))
= 5.5683279... * 1.8477590650.. / ( 4 * 1.43451884..... * 1.0896523574...).
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• H. S. Wrigge, An Elliptic Integral Identity, Math. Comp. 27 (1973) no 124, p 839.
• I. J. Zucker and G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Camb. Phil. Soc. 131 (2001) 309-319 (2.4)

Programs

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

Maple

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

Mathematica

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 *)
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 *)

PARI

default(realprecision, 100); Pi^(3/2)*sqrt(2 + sqrt(2))/(4* gamma(5/8)*gamma(7/8)) \\ G. C. Greubel, Sep 27 2018

PARI

ellK(sqrt(2)-1) \\ Charles R Greathouse IV, Feb 04 2025

Extensions

More terms from Robert G. Wilson v, Jul 19 2007