OFFSET
0,1
COMMENTS
The other (complex) roots are w1*((1 + (1/9)*sqrt(129))/4)^(1/3) + ((1 - (1/9)*sqrt(129))/4)^(1/3) = -0.2119268995... + 1.0652413023...*i, and its complex conjugate, where w1 = (-1 + sqrt(3))/2 = exp((2/3)*Pi*i).
Using hyperbolic functions these roots are -(1/3)*sqrt(3)*(sinh((1/3)*arcsinh((3/4)*sqrt(3))) - sqrt(3)*cosh((1/3)*arcsinh((3/4)*sqrt(3)))*i), and its complex conjugate.
FORMULA
r = ((1 +(1/9)*sqrt(129))/4)^(1/3) - (1/3)*((1 + (1/9)*sqrt(129))/4)^(-1/3).
r = ((1 + (1/9)*sqrt(129))/4)^(1/3) + w1*((1 - (1/9)*sqrt(129))/4)^(1/3), where w1 = (-1 + sqrt(3))/2, one of the complex roots of x^3 - 1.
r = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).
EXAMPLE
0.423853799069783271378040062625515233676388197185177540823008396819954728...
MATHEMATICA
RealDigits[x /. FindRoot[2*x^3 + 2*x - 1, {x, 1}, WorkingPrecision -> 100]][[1]] (* Amiram Eldar, Sep 29 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 29 2022
STATUS
approved