A138283 Decimal expansion of the imaginary part of z0, the smallest nonzero first-quadrant solution of z = Sin(z).

2, 7, 6, 8, 6, 7, 8, 2, 8, 2, 9, 8, 7, 3, 2, 1, 5, 3, 2, 4, 9, 5, 3, 1, 3, 9, 1, 3, 0, 5, 1, 4, 0, 9, 0, 2, 0, 7, 1, 5, 5, 5, 1, 6, 5, 9, 0, 0, 0, 8, 6, 5, 6, 7, 4, 7, 3, 1, 3, 0, 3, 8, 4, 2, 8, 4, 8, 0, 2, 6, 7, 1, 4, 2, 7, 6, 9, 6, 3, 8, 1, 3, 0, 8, 4, 8, 0, 1, 6, 0, 3, 8, 0, 8, 1, 7, 0, 2, 4, 3, 0, 5, 8, 5, 6

Offset

1,1

Comments

z0 is a repelling fixed point of Sin(z). The constant z0 appears in the paper by Shenderov.

From Jianing Song, Oct 11 2021: (Start)

Positive root of cosh(y) * sqrt(1 - (y/sinh(y))^2) - arccos(y/sinh(y)) = 2*Pi.

In general, all roots of sin(z) = z are given by z = 0 and z = +-(x_k)+-(y_k)*i, where x_k is the root of cos(x) * sqrt((x/sin(x))^2 - 1) - arccosh(x/sin(x)) = 0 in the range (2*k*Pi, (2*k+1/2)*Pi), y_k is the positive root of cosh(y) * sqrt(1 - (y/sinh(y))^2) - arccos(y/sinh(y)) = 2*k*Pi. (End)

Links

Table of n, a(n) for n=1..105.

E. L. Shenderov, Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls, Acoustical Physics 46 (2000), 357-363.

Example

2.768678282987321532495314...

Mathematica

z0 = FindRoot[{Re[Sin[x+I*y]]==x, Im[Sin[x+I*y]]==y}, {{x, 7}, {y, 3}}, WorkingPrecision->150]; RealDigits[z0[[2, 2]]]

Prog

(PARI) solve(y=2.7, 2.8, cosh(y) * sqrt(1 - (y/sinh(y))^2) - acos(y/sinh(y)) - 2*Pi) \\ Jianing Song, Oct 11 2021

Crossrefs

Cf. A138282 (real part), A348297 (real part of the second nontrivial root), A348298 (imaginary part of the second nontrivial root).

Sequence in context: A158241 A156591 A233770 * A308682 A117968 A217568

Adjacent sequences: A138280 A138281 A138282 * A138284 A138285 A138286

Keyword

cons,nonn

Author

T. D. Noe, Mar 12 2008

Status

approved