Complex Analysis - Maths KU

ar
Figure 2.12 Magnification
C
C′
M
M(z)
4 6
Figure 2.13 Image of a circle under
magnification
2
2.3 Linear Mappings 71
The product ar in (5) is a real number since both a and r are real numbers,
and from this it follows that the magnitude of M(z) isar. Assume that
a>1. Then from (5) we have that the complex points z and M(z) have
the same argument θ but different moduli r �= ar. If we plot both z and
M(z) in the same copyof the complex plane, then M(z) is the unique point
on the rayemanating from 0 and containing z whose distance from 0 is ar.
Since a>1, M(z) isa times farther from the origin than z. Thus, the linear
mapping M(z) =az can be visualized in a single copyof the complex plane
as the process of magnifying the modulus of the point z bya factor of a to
obtain the point M(z). See Figure 2.12. The real number a is called the
magnification factor of M. If 0