A138282 Decimal expansion of the real part of z0, the smallest nonzero first-quadrant solution of z = Sin(z).

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

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)

Real root of cos(x) * sqrt((x/sin(x))^2 - 1) - arccosh(x/sin(x)) = 0 in the range (2*Pi, 2.5*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

7.497676277776385498272325...

Mathematica

z0 = FindRoot[{Re[Sin[x+I*y]]==x, Im[Sin[x+I*y]]==y}, {{x, 7}, {y, 3}}, WorkingPrecision->150]; RealDigits[z0[[1, 2]]]

Prog

(PARI) solve(x=7.4, 7.5, cos(x) * sqrt((x/sin(x))^2 - 1) - log(x/sin(x) + sqrt((x/sin(x))^2 - 1))) \\ Jianing Song, Oct 11 2021

Crossrefs

Cf. A138283 (imaginary part), A348297 (real part of the second nontrivial root), A348298 (imaginary part of the second nontrivial root).

Sequence in context: A199448 A110644 A117028 * A198572 A155823 A195452

Adjacent sequences: A138279 A138280 A138281 * A138283 A138284 A138285

Keyword

cons,nonn

Author

T. D. Noe, Mar 12 2008

Status

approved