OFFSET
0,1
COMMENTS
The other (complex) roots are w1*(1/2 + (1/36)*sqrt(330))^(1/3) + (1/2 - (1/36)*sqrt(330))^(1/3) = -0.4175611742... + 1.0114702183....*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp((2/3)*Pi*i) is one of the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (1/6)*sqrt(6)*(-cosh((1/3)*arccosh(3*sqrt(6))) + sqrt(3)*sinh((1/3)*arccosh(3*sqrt(6)))*i), and its complex conjugate.
FORMULA
r = ((108 + 6*sqrt(330))^(1/3) - 6*(108 + 6*sqrt(330))^(-1/3))/6.
r = (6*(1/2 + (1/36)*sqrt(330))^(1/3) - (1/2 + (1/36)*sqrt(330))^(-1/3))/6.
r = (1/2 + (1/36)*sqrt(330))^(1/3) + w1*(1/2 - (1/36)*sqrt(330))^(1/3), where w1 = (-1 = sqrt(3)*i)/2 is one of the complex roots of x^3 - 1.
r= (1/3)*sqrt(6)*sinh((1/3)*arcsinh(3*sqrt(6))).
EXAMPLE
0.835122348481366514291620038596702299165411487780433601936279731538589...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 + x - 2, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 29 2022
STATUS
approved