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A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

%I A173201 #42 May 03 2025 09:36:06
%S A173201 1,9,0,5,6,9,5,7,2,9,3,0,9,8,8,3,8,9,4,8,8,2,6,6,6,4,3,7,1,6,0,9,6,6,
%T A173201 7,0,3,4,9,5,0,4,3,1,2,1,6,1,2,8,0,3,2,1,2,1,9,3,5,6,4,5,5,9,9,9,4,5,
%U A173201 4,4,2,4,0,9,9,5,7,9,5,0,2,2,7,5,7,1,6,1,6,6,3,4,6,4,6,3,0,3,9,7,1,5,3,9,8
%N A173201 Fixed point of the iteration x |--> x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).
%C A173201 Decimal expansion of psi, the unique solution on (0,Pi) of sin(psi) - psi*cos(psi) = Pi/2, an auxiliary constant used in the Hall-Tenenbaum inequality applied to real multiplicative functions. - _Jean-François Alcover_, Sep 05 2014
%H A173201 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 28.
%H A173201 M. Fraser, <a href="http://www.jstor.org/stable/2690163">A tale of two goats</a>, Math. Mag., 55 (1982), 221-227.
%H A173201 Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Roemisch_JAVA.htm#ZZZZZ0004">Iterationsrechner mit Algorithmus</a>
%H A173201 Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/langeZahlen/19056957293.pdf">10000 digits</a>
%H A173201 Gerd Lamprecht, <a href="http://www.gerdlamprecht.de/Zahlenfolgen.html">Zahlenfolgen (sequence)</a>
%F A173201 x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).
%F A173201 Equals 2*arccos(A133731/2).
%e A173201 A133731 = cos(1.9056957293.../2)*2.
%t A173201 psi = x /. FindRoot[Sin[x] - x*Cos[x] == Pi/2, {x, 2}, WorkingPrecision -> 102]; RealDigits[psi] // First (* _Jean-François Alcover_, Sep 05 2014 *)
%o A173201 (Gerd Lamprecht Iterationsrechner) #(sin(x)-cos(x)*x-PI/2)/(sin(x)*x)@Na=2;@Nb=a;a=a-Fx(a);@N@Aa-b)%3C%204e-16@N0@N1@Nc=cos(a/2)*2;
%o A173201 (PARI) solve(x=1,2,sin(x)-x*cos(x)-Pi/2) \\ _Charles R Greathouse IV_, Mar 03 2021
%Y A173201 Cf. A072112, A133731.
%K A173201 cons,nonn
%O A173201 1,2
%A A173201 Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010