Complex Analysis - Maths KU

3i
2i
i
S
z
1 2 3
S′
T(z)
Figure 2.9 Image of a square under
translation
2.3 Linear Mappings 69
representation of the complex number b, the mapping T (z) =z + b is also
called a translation by b.
EXAMPLE 1 Image of a Square under Translation
Find the image S ′ of the square S with vertices at 1 + i, 2+i, 2+2i, and
1+2i under the linear mapping T (z) =z +2− i.
Solution We will represent S and S ′ in the same copyof the complex plane.
The mapping T is a translation, and so S ′ can be determined as follows.
Identifying b = x0 + iy0 = 2 + i(−1) in (1), we plot the vector (2, −1)
originating at each point in S. See Figure 2.9. The set of terminal points of
these vectors is S ′ , the image of S under T . Inspection of Figure 2.9 indicates
that S ′ is a square with vertices at:
T (1 + i) = (1 + i)+(2− i) =3 T (2 + i) =(2+i)+(2− i) =4
T (2+2i) =(2+2i)+(2− i) =4+i T(1+2i) =(1+2i)+(2− i) =3+i.
Therefore, the square S shown in color in Figure 2.9 is mapped onto the square
S ′ shown in black bythe translation T (z) =z +2− i.
From our geometric description, it is clear that a translation does not
change the shape or size of a figure in the complex plane. That is, the image
of a line, circle, or triangle under a translation will also be a line, circle, or
triangle, respectively. See Problems 23 and 24 in Exercises 2.3. A mapping
with this propertyis sometimes called a rigid motion.
Rotations A complex linear function
R(z) =az, |a| =1, (2)
is called a rotation. Although it mayseem that the requirement |a| = 1 is a
major restriction in (2), it is not. Keep in mind that the constant a in (2) is
a complex constant. If α is anynonzero complex number, then a = α/ |α| is
a complex number for which |a| = 1. So, for anynonzero complex number α,
we have that R(z) = α
z is a rotation.
|α|
Consider the rotation R given by(2) and, for the moment, assume that
Arg(a) > 0. Since |a| = 1 and Arg(a) > 0, we can write a in exponential form
as a = e iθ with 0