Complex Linear Functions

The function is w = az+b. The red grid is in the z = x+iy domain, the blue image is in the w = u+iv domain.

We will consider w = (1-i) z+(2+i).


Along the real axis, think how w = az+b is
(1-i) t + (2+i) = [t+2, -t+1]
which yields the path y = -x + 3.


However, along the imaginary axis, w = az+b is
(1-i) it + (2+i) = [t+2, t+1]
which yields the path y = x - 1.

An alternative is to consider linear maps as the composition translation(magnification(rotation)). The linear function
w = (1+31/2i) z + (2-i)
is equal to
z1 = eip/3 z    (rotation)
z2 = 2 z1    (magnificaton)
w = z2 + (2-i)    (translation)
This decomposition is shown acting on z = (-0.5-0.5i)..(0.5+0.5i) in the figure below.

[Maple Plot