OFFSET
0,1
COMMENTS
The other real root is -A358190. One of the complex roots is 0.4603551884... + 1.1393176803...*i, given by (sqrt(2)*u + sqrt(-2*u^2 - 2*sqrt(2*u)))/(2*sqrt(u)), where u is given in the formula below. The other complex root is its conjugate (with a minus sign for the second sqrt).
FORMULA
r = (-sqrt(2)*u + sqrt(-2*u^2 + 2*sqrt(2*u)))/(2*sqrt(u1)), where u = (((3/4)*(9 + sqrt(129)))^(1/3) + w1*((3/4)*(9 - sqrt(129)))^(1/3))/3 = 0.4238537990..., and w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3), is one of the complex roots of x^3 - 1. Alternatively u = (2/3)*sqrt(3)*sinh((1/3)*arcsinh((3/4)*sqrt(3))).
EXAMPLE
0.47462661756260555032941320989493141266736136591947852234956563261143...
MATHEMATICA
RealDigits[x /. FindRoot[x^4 + 2*x - 1, {x, 1}, WorkingPrecision -> 120], 10, 120][[1]] (* Amiram Eldar, Dec 07 2022 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Dec 07 2022
STATUS
approved