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A348297
Decimal expansion of real part of the second nontrivial root of sin(z) = z in the first quadrant.
3
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
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.
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
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Oct 10 2021
STATUS
approved