Cubic critical points

For f(z) = z3 + Az + C,

f'(z) = 3z2 + A

The critical points of f(z) are the solutions of f'(z) = 0,

so z = sqrt(-A/3) and z = -sqrt(-A/3) are the critical points of f(z) = z3 + Az + C.

Writing A = a + i*b, after some simplification we find if b <= 0

sqrt(-A/3) = ((Sqrt[Sqrt[a2 + b2]]/Sqrt[6])*Sqrt[1 - (a/Sqrt[a2 + b2])]) + i*((Sqrt[Sqrt[a2 + b2]]/Sqrt[6])*Sqrt[1 + (a/Sqrt[a2 + b2])])

and if b > 0

sqrt(-A/3) = ((Sqrt[Sqrt[a2 + b2]]/Sqrt[6])*Sqrt[1 - (a/Sqrt[a2 + b2])]) - i*((Sqrt[Sqrt[a2 + b2]]/Sqrt[6])*Sqrt[1 + (a/Sqrt[a2 + b2])])