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%I #14 Jun 24 2021 07:31:11
%S 8,1,2,8,2,5,2,4,2,1,0,5,5,0,7,2,6,0,7,6,0,0,8,7,1,2,3,1,1,8,3,7,0,2,
%T 9,8,6,4,7,0,1,0,1,3,4,0,5,2,8,7,0,3,4,0,6,5,7,3,6,0,0,3,9,5,8,0,7,2,
%U 7,4,7,2,6,7,9,4,0,2,2,7,2,3,8,3,9,1,2,5,2,9,4,7,9,0,9,6,4,6,7,2,9,8,2
%N Decimal expansion of 'mu', a Young-Fejér-Jackson constant linked to the positivity of certain sine sums.
%C Named after the English mathematician William Henry Young (1863-1942), the Hungarian mathematician Lipót Fejér (or Leopold Fejér, 1880-1959) and the American mathematician Dunham Jackson (1888-1946). - _Amiram Eldar_, Jun 24 2021
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 242.
%H Leopold Fejér, <a href="https://doi.org/10.1515/crll.1910.138.22">Lebesguessche Konstanten und divergente Fourierreihen</a>, J. Reine Angew. Math., Vol.. 138 (1910), pp. 22-53; <a href="https://eudml.org/doc/149327">alternative link</a>.
%H Dunham Jackson, <a href="https://doi.org/10.1007/BF03014798">Über eine trigonometrische Summe</a>, Rendiconti del Circolo Matematico di Palermo (1884-1940), Vol. 32, No. 1 (1911), pp. 257-262.
%H W. H. Young, <a href="https://doi.org/10.1112/plms/s2-11.1.357">On a certain series of Fourier</a>, Proc. London Math. Soc., Vol. 11 (1912), pp. 357-366.
%F Given lambda from A227423, mu is the unique positive solution of (1+lambda)*sin(mu*Pi) = mu*sin(lambda*Pi).
%e 0.81282524210550726076008712311837029864701013405287034065736003958072747...
%p Digits:= 127:
%p lambda:= solve((1+x)*Pi - tan(x*Pi), x):
%p mu:= fsolve((1+lambda)*sin(x*Pi)-x*sin(lambda*Pi), x, 0.1..1):
%p s:= convert(mu, string):
%p seq(parse(s[n+1]), n=1..Digits-10); # _Alois P. Heinz_, Jul 11 2013
%t mu /. FindRoot[(1 + lambda)*Pi == Tan[lambda*Pi] && (1 + lambda)*Sin[mu*Pi] == mu* Sin[lambda*Pi], {lambda, 2/5}, {mu, 4/5}, WorkingPrecision -> 100] // RealDigits // First
%Y Cf. A227422, A227423, A227425.
%K nonn,cons
%O 0,1
%A _Jean-François Alcover_, Jul 11 2013
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