A348297 - OEIS

A348297

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

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

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OFFSET

2,2

COMMENTS

Real root of cos(x) * sqrt((x/sin(x))^2 - 1) - arccosh(x/sin(x)) = 0 in the range (4*Pi, 4.5*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.

LINKS

Table of n, a(n) for n=2..91.

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[x /. 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(x=13.8, 13.9, cos(x) * sqrt((x/sin(x))^2 - 1) - log(x/sin(x) + sqrt((x/sin(x))^2 - 1)))

CROSSREFS

Cf. A348298 (imaginary part), A138282 (real part of the first nontrivial root), A138283 (imaginary part of the first nontrivial root).

Sequence in context: A197040 A217870 A154927 * A363955 A220789 A276006

Adjacent sequences: A348294 A348295 A348296 * A348298 A348299 A348300

KEYWORD

nonn,cons

AUTHOR

Jianing Song, Oct 10 2021

STATUS

approved