A348298 Decimal expansion of imaginary 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

Offset

1,1

Comments

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) - 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.

Links

Table of n, a(n) for n=1..100.

Example

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}.

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A105104 A229087 A142961 * A101777 A204154 A267089

Adjacent sequences: A348295 A348296 A348297 * A348299 A348300 A348301

Keyword

nonn,cons

Author

Jianing Song, Oct 10 2021

Status

approved