The Inverse of the Complex Cubing Rrelation

For the cubing function, the inverse relation is the cube root relation z = w1/3. In the case of the real cubing function [LINK], the inverse relation is a function, since every real number has a unique real cube root. On the other hand, every complex number (other than zero) has three distinct cube roots.


Notice that the uv-plane is the vertical plane on the left of the cube (the u axis is blue and the v axis is white). The picture on the left represents the real part of the cube root (the red axis is for x), and on the right is the imaginary part (the green axis is for y). The fact that any non-zero complex number has three roots is apparent from the fact that a line parallel to the red or green axis will cut the surface in three points (that is, at three different x or y values).