Functions of a complex variable in four
dimensions
A function of a complex variable represents a
2-manifold embedded in four dimensions. Here, the
white grid is the z=x+iy complex plane. The
function w=exp(z), or u+iv=exp(x+iy) is projected
from four-dimensional (x,y,u,v) space to the
screen, in a similar same way a 3D image is
projected onto a 2D picture. This projection shows
the imaginary part of exp(x+iy). Note that
exp(x+iy)=exp(x)*(cos(y)+i*sin(y)). One can see
both the sinusoidal variation with y and the
exponential variation with x in this view.
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