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23.2 Construction of Conformal Mappings 767

v

y 3i
w = 3z
i
1 3
x u

FIGURE 23.21 Mapping |z| < 1 onto |w| < 3 in
Example 23.10.

EXAMPLE 23.10
We will map the open unit disk D :|z| < 1 conformally onto the open disk D :|w| < 3.
∗
Clearly this is just a magnification, so w = f (z) = 3z will do, expanding the unit disk to a
disk of radius 3 while leaving the origin at the center (Figure 23.21). Observe that f carries the
boundary of D onto the boundary of D .
∗

EXAMPLE 23.11
Map the open unit disk conformally onto the domain D :|w| > 3.
∗
Here we are mapping the open unit disk to the complementary domain of the preceding
example. We know that f (z) = 3z maps D conformally onto |w| < 3. We also know that inver-
sions map the domains interior to circles to domains exterior to circles. Thus, combine this map
with an inversion, letting
3
g(z) = f (1/z) = .
z
This maps |z| < 1 onto |w| > 3 (Figure 23.22). Again observe that g maps the boundary to the
boundary.

w = 3/z
v

y 3i

i
3
1 x u

FIGURE 23.22 Mapping |z| onto |w| > 3 in Example 23.11.

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