|
%I #25 Oct 02 2023 02:38:36
%S 9,2,8,9,0,2,5,4,9,1,9,2,0,8,1,8,9,1,8,7,5,5,4,4,9,4,3,5,9,5,1,7,4,5,
%T 0,6,1,0,3,1,6,9,4,8,6,7,7,5,0,1,2,4,4,0,8,2,3,9,7,0,0,6,1,4,2,1,7,2,
%U 9,3,7,5,2,4,7,2,8,6,5,0,7,0,7,0,5,2,4,1,5,8,7,0,6,1,4,2,4,7,1,4,4
%N Decimal expansion of Varga's constant.
%C Equals the reciprocal of the one-ninth constant A072558.
%C Named after the American mathematician Richard Steven Varga (1928-2022). - _Amiram Eldar_, Jun 22 2021
%D R. S. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 60, Philadelphia, PA: SIAM, 1990. See Chapter 2, pp. 23-38.
%H G. C. Greubel, <a href="/A073007/b073007.txt">Table of n, a(n) for n = 1..10000</a>
%H A. J. Carpenter, A. Ruttan and R. S. Varga, <a href="https://doi.org/10.1007/BFb0072427">Extended numerical computations on the "1/9" conjecture in rational approximation theory</a>, in: P. R. Graves-Morris, E. B. Saff and R. S. Varga (eds.), Rational approximation and interpolation, Springer, Berlin, Heidelberg, 1984, pp. 383-411; <a href="http://www.math.kent.edu/~varga/pub/paper_148.pdf">alternative link</a>.
%H Alphonse P. Magnus and Jean Meinguet, <a href="https://doi.org/10.1023/A:1019141226189">The elliptic functions and integrals of the '1/9' problem</a>, Numerical Algorithms, Vol. 24, No. 1 (2000), pp. 117-139; <a href="https://perso.uclouvain.be/alphonse.magnus/onine/antw99p.pdf">alternative link</a>.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap91.html">One-ninth constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/One-NinthConstant.html">One-Ninth Constant</a>.
%e 9.28902549192081891875544943595174506...
%t nmax=250; c = k /. FindRoot[EllipticK[k^2] == 2*EllipticE[k^2], {k, 9/10}, WorkingPrecision -> nmax]; Take[RealDigits[1/N[Exp[-Pi*(EllipticK[1 - c^2]/EllipticK[c^2])], nmax]][[1]], 200] (* _G. C. Greubel_, Mar 10 2018 *)
%t RealDigits[v /. FindRoot[4 EllipticE[InverseEllipticNomeQ[1/v]] == Pi EllipticTheta[3, 0, 1/v]^2, {v, 9, 9, 10}, WorkingPrecision -> 101]][[1]] (* _Jan Mangaldan_, Jun 25 2020 *)
%Y Cf. A072558.
%K cons,nonn
%O 1,1
%A _Robert G. Wilson v_, Aug 03 2002
|
