A138282 - OEIS

%I #11 Oct 16 2021 06:13:15

%S 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,

%T 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,

%U 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

%N Decimal expansion of the real 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 Real root of cos(x) * sqrt((x/sin(x))^2 - 1) - arccosh(x/sin(x)) = 0 in the range (2*Pi, 2.5*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 7.497676277776385498272325...

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

%o (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

%Y Cf. A138283 (imaginary 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