Is there a general equation for a cube root of a complex number that does not exploit De Moivre's Theorem or in any way use trigonometric functions?
For example, a square root of a complex number $x$ is $$\sqrt{\frac{|x|+\operatorname{Re}(x)}{2}}+i\sqrt{\frac{|x|-\operatorname{Re}(x)}{2}}.$$ Is there a similar equation for a cube root of $x$?
By introducing $a$ and $b$ such that $(a+ib)^3=x$, we can then expand and obtain two equations in $a$ and $b$ by equating the real and imaginary parts on each side.
However, the resulting equations are cubic and I don't know any method to find the roots of a cubic equation without having to take the cube root of a complex number, which is the problem I want to solve in the first place.
Best Answer
There is no such nice formula for the cube root of a complex number with both real and imaginary parts nonzero. If you write out the real and imaginary parts of your cube root, you wind up solving cubic equations in one variable that have three irrational roots. This is the Casus Irreducibilis http://en.wikipedia.org/wiki/Casus_irreducibilis
In turn, for each of those cubics, Cardano's method leads you to finding the cube roots of other complex numbers. All very circular, and never gets anywhere.