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

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, 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, 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

Offset: 1

Author

Gerd Lamprecht (gerdlamprecht(AT)googlemail.com), Feb 12 2010

Keywords

• cons
• nonn

Comments

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

Examples

			A133731 = cos(1.9056957293.../2)*2.
		

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 28.
• M. Fraser, A tale of two goats, Math. Mag., 55 (1982), 221-227.
• Gerd Lamprecht, Iterationsrechner mit Algorithmus
• Gerd Lamprecht, 10000 digits
• Gerd Lamprecht, Zahlenfolgen (sequence)

Crossrefs

Cf. A072112, A133731.

Programs

Mathematica

psi = x /. FindRoot[Sin[x] - x*Cos[x] == Pi/2, {x, 2}, WorkingPrecision -> 102]; RealDigits[psi] // First (* Jean-François Alcover, Sep 05 2014 *)

PARI

solve(x=1,2,sin(x)-x*cos(x)-Pi/2) \\ Charles R Greathouse IV, Mar 03 2021

Formula

x := x - (sin(x) - cos(x)*x - Pi/2)/(sin(x)*x).

Equals 2*arccos(A133731/2).