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Imaginary Decimal expansion of imaginary part of the second nontrivial root of sin(z) = z in the first quadrant.

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RealDigits[y /. FindRoot[{Re[Sin[x + I*y]] == x, Im[Sin[x + I*y]] == y}, {{x, 14}, {y, 3}}, WorkingPrecision -> 100], 10, 90][[1]] (* Amiram Eldar, Oct 10 2021 *)

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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) - log(x/sin(x) + sqrt(arccosh(x/sin(x))^2 - 1)) = 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.

z = 13.8999597139... + 3.3522098848...*i is the unique root of sin(z) = z in the region {z: 4*Pi <= Re(z) < 6*Pi, Im(z) >= 0}.

Cf. A348297 (real part), A138282 (real part of the first nontrivial root), A138283 (imaginary part of the first nontrivial root).

allocated for Jianing SongImaginary part of the second nontrivial root of sin(z) = z in the first quadrant.

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

1,1

Positive root of cosh(y) * sqrt(1 - (y/sinh(y))^2) - arccos(y/sinh(y)) = 4*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) - log(x/sin(x) + sqrt((x/sin(x))^2 - 1)) = 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.

(PARI) solve(y=3.3, 3.4, cosh(y) * sqrt(1 - (y/sinh(y))^2) - acos(y/sinh(y)) - 4*Pi)

Cf. A138283 (imaginary part of the first nontrivial root).

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nonn,cons

Jianing Song, Oct 10 2021

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allocated for Jianing Song

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