1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

y

Z=l+z\-._.

1- z

10 .3 • MAPPINGS INVOLVING E LEMENTARY FUNCTIONS 413

w=(~1- z ) 2

• u

Figure 10.12 The composite transformat ion w = ( ~)

2

.

Solution The function w = f (z) is the composition of the bilinear transfor-

mation Z = ~ followed by the mapping w = Z^2 • The image of the half-disk

under the bilinear mapping Z =
1

1

+ z is the first quadrant X > 0, Y > O; the

• z

image of the segment y = 0, -1<x<1, is the positive X -axis; and the image

of the semicircle is the positive Y-axis. The mapping w = Z^2 then maps the
first quadrant in the Z plane onto the upper half-plane Im (w) > 0, as shown in
Figure 10.12.

• EXAMPLE 10. 11 Consider the function w = f (z) = (z^2 - 1) t, which is

the composition of the functions Z = z^2 - 1 and w = Z~, where the branch of

the square root is Z~ = R~(cos! + isin !), where 0:::; IP< 211'. Show that the

transformation w = f (z) maps the upper half-plane Im (z) > 0 one-to-one and
onto the upper half-plane Im (w) > 0 slit along the segment u = 0, 0 < v:::; 1.

Solution The function Z = z^2 - 1 maps the upper half-plane Im(z) > 0

one-to-one and onto the Z-plane slit along the ray Y = 0, X ;::=: -1. Then the

function w = Z~ maps the slit plane onto the slit half-plane, as shown in Figure

10.13.

Remark 10. 1 The images of the horizontal lines y = b are curves in the

w plane that bend around the segment from 0 to i. The curves represent the
streamlines of a fluid flowing across the w plane. We discuss fluid flows in more
detail in Section 11.7. •