Complex Integer Power Functions

The function is w = z n. The blue grid is in the z = x+iy domain, the red image is in the w = u+iv domain.
First we consider w = z2.

Along lines parallel to the real axis, z2 behaves like its real counterpart.

As we travel up the imaginary axis, the image stays on the negative real axis since i2 = -1.

Now, look at w = z3.

As we are coming to expect, along the real axis, the function behaves as its real counterpart.

The behaviour along the imaginary axis is somewhat unusual at first glance. The easiest way to understand these functions is to write the expressions in polar form. Then w = z3 becomes
z 3 = r 3e 3iq
We see that the modulus is cubed and the argument (angle) is tripled.