Complex Integer Power Functions
The function is w = z n. The red grid is in the
z = x+iy domain, the blue 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 3e3iq
We see that the modulus is cubed and the argument (angle) is tripled.