The Complex Squaring Function (Polar Coordinates)
Consider the graph of the squaring function, w =
z2, where the domain is restricted to the unit
disc. When we look at the graph itself, we represent the domain in the
(x,y)-plane since we write z = x +
yi. Writing w = u + vi, we
can use the third dimension to represent the real part of w, (the
u value), the imaginary part (the v value), or some
combination of the two.
If we use the equation cos(a) u + sin(a)
v as the third dimension, then as a varies from 0 to
pi/2, we will move from the real part to the imaginary part of the
function. Effectively, we are projecting the four-dimensional graph into
three-space and rotating the graph so that we can see it from different
viewpoints. The view on the left below reprents the real-part of the
graph, and on the right, the imaginary part of the complex squaring function.
The movie shows the rotation that moves between the two.
Notice that the red and green axes remain the same (these are the x
and y axes) while the third axis goes from blue to white,
indicating the change from seeing the u axis to seeing the v
axis.
