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772 CHAPTER 23 Conformal Mappings and Applications

y
2i
i
2 1
x

v
y
*
D : u + v > 0 v
u + v > 0
2i
2 u
u
x
D : z < 2

FIGURE 23.27 The domains |z| < 2 and u + v> 0 FIGURE 23.28 Mapping |z| < 2 onto the rotated
in Example 23.16. half-plane u + v> 0 in Example 23.16.

As usual, first look for mappings we already have that relate to this problem. We can map
|z| < 2 to the unit disk |Z| < 1 (multiply by 1/2). We also know a mapping from the unit disk
to the right half-plane, say from the Z-plane to the W-plane. Finally, we can obtain D from the
∗
right half-plane by a counterclockwise rotation through π/4 radians, which is an effect achieved
by multiplying by e iπ/4 (a straight rotation). This suggests that we construct the mapping we want
in the stages shown in Figure 23.28:
|z| < 2 →|Z| < 1 → Re(W)> 0 → u + v> 0.
For the first step, let
1
Z = z.
2
Next, following Example 23.13, we write a mapping between the unit circle and the right half-
plane. In the current notation for the planes, this is
i − Z
W = .
i + Z
Finally, we want to rotate the effect of this mapping counterclockwise by π/4 (last stage of
Figure 23.28). For this, set
w = We iπ/4 .
Putting everything together, a conformal mapping from D to D is given by
∗

i − Z
w = We iπ/4 = e iπ/4
i + Z
i − z/2 iπ/4

= e
i + z/2

2i − z iπ/4
= e = f (z).
2i + z

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October 14, 2010 15:39 THM/NEIL Page-772 27410_23_ch23_p751-788