Figure 12-10. Representing function of complex variable as mapping.
Polar coordinates (r, θ)in the z−plane correspond to polar coordinates (R, φ )in
the ω−plane, where x=rcos θ, y =rsin θin the z−plane and u=Rcos φ, v =Rsin φin
the ω−plane. A curve y=G(x)in the z−plane has the image curve with parametric
form
u=u(x, G (x)), v =v(x, G (x))
in the ω−plane, which produces the image curve v=F(u).
Functions of a complex variable ω=f(z)represent a mapping from the z−plane
to the ω−plane. One usually selects special regions Sand curves Cto illustrate the
mappings. For example, circles, squares, triangles, etc.
