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768 CHAPTER 23 Conformal Mappings and Applications

Y
3i
Z = 3z
y

x 3 X

v
w = 3z + i 4i

w = Z + i
i

u
−2i

FIGURE 23.23 Mapping |z| < 1 onto |w − i| < 3 in Exam-
ple 23.12.

EXAMPLE 23.12
Map the open unit disk onto the disk |w − i| < 3 or radius 3 centered at i in the w-plane.
Figure 23.23 suggests one way to construct this mapping. We want to expand the unit disk’s
radius by a factor of 3, then translate the resulting disk up one unit. Put an intermediary Z =
X + iY plane between the z-plane and the final w-plane, and map in steps:
z → Z = 3z → w = Z + i = 3z + i = w = f (z).
Note that the boundaries map to each other: the unit circle |z|= 1 maps to

|w − i|= 3|z|= 3,
which is the circle of radius 3 about i.

EXAMPLE 23.13
We will find a conformal mapping of the right half-plane Re(z)> 0 to the unit disk |w| < 1.
Let S denote the right half-plane in the z-plane, and K denote the unit disk in the w-plane
(Figure 23.24). The boundary of S is the imaginary axis, and the boundary of K is the unit circle.
A bilinear transformation may work here, since the boundaries are a line and a circle. Pick three
points on the imaginary axis (boundary of S) in order down the axis for positive orientation of
this axis as the boundary of the right half-plane (walking in this direction, the right half-plane is
over our left shoulder). We will use z 1 = i, z 2 = 0, and z 3 =−i, although other choices will do.
Now choose three image points in order counterclockwise (positive orientation) on the unit circle
in the w-plane, say w 1 = 1,w 2 = i, and w 3 =−1. This is the direction we have to walk around
the unit circle to have the unit disk over our left shoulder.

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