1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.1 • FUNCTIONS AND LINEAR MAPPINGS 59

(Method 2): When we write w = f (z) in Cartesian form as

w = u + iv = i(x +iy) + i = -y+i(x+ 1),

we see that the transformation can be given by the equations u = - y and

v = x + 1. Substituting x = v - 1 in -the jnequality Re (z) = x 2 1 gives

v -121, or v 2 2, which is the upper half-plane Im (w) 2 2.

(Method 3): The effect of the transformation w = f (z) is a rotation of the plane

through the angle °' = ~ (when z is multiplied by i) followed by a translation

by the vector B = i. The first operation yields the set Im ( w) 2 1. The second

shifts this set up 1 unit, resulting in the set Im ( w) 2 2.

We illustrate this result in Figure 2.9.

y v

w= it+i

Figure 2.9 The linear transformation w = f (z) = iz + i.

Translations and rotat ions preserve angles. First, magnifications rescale dis-

tance by a factor K , so it follows that triangles are mapped onto similar triangles,

preserving angles. Then, because a linear transformation can be considered to

be a composition of a rotation, a magnification, and a translation, it follows

that linear transformations preserve angles. Consequently, any geometric object

is mapped onto an object that is similar to the original object; hence linear

transformations can be called similarity mappings.

• EXAMPLE 2.10 Show that the image of D 1 (-1 - i) = {z: lz + 1 + ii< l}

under the transformation w = (3 - 4i) z + 6 + 2i is the open disk Ds (- 1 + 3i) =

{w: lw + 1 - 3il < 5}.