A138283 - OEIS

%I #11 Oct 16 2021 06:13:08

%S 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,

%T 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,

%U 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

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

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

%C From _Jianing Song_, Oct 11 2021: (Start)

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

%C 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)

%H E. L. Shenderov, <a href="https://doi.org/10.1134/1.29892">Helmholtz equation solutions corresponding to multiple roots of the dispersion equation for a waveguide with impedance walls</a>, Acoustical Physics 46 (2000), 357-363.

%e 2.768678282987321532495314...

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

%o (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

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

%K cons,nonn

%O 1,1

%A _T. D. Noe_, Mar 12 2008