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760 CHAPTER 23 Conformal Mappings and Applications

1/ z
i

z

1

1/z
FIGURE 23.15 Image of z under an inver-
sion in Example 23.6.

and
arg(w) = arg(1) − arg(z) =−arg(z)
(within integer multiples of 2π). This means that we arrive at T (z) by moving 1/|z| units from the
origin along the line from 0 to z and then reflecting this point across the real axis (Figure 23.15).
Points enclosed by the unit circle map outside this circle, and points exterior to the unit circle
map to the inside. Points on the unit circle remain on this circle but are moved around it, except
√
for 1 and −1, which remain fixed under T . For example, T maps (1+i)/ 2, which has argument
√
π/4, to (1 − i)/ 2, which is still on the unit circle but has argument −π/4.
We will now show that translations, rotation/magnifications, and inversions are the funda-
mental bilinear transformations in the sense that the effect of any bilinear transformation can be
achieved as a sequence of mappings of these three types. To see how to do this, begin with
az + b
T (z) = .
cz + d
If c = 0, then
a b
T (z) = z + ,
d d
which is a rotation/magnification followed by a translation:
rot/mag a trans b
z −−−→ z −→ + .
d d
If c = 0, then T is the result of the following sequence:
rot/mag trans
z −−−→ cz −→ cz + d
1 rot/mag bc − ad 1
inv
−→ −−−→
cz + d c cz + d
trans bc − ad 1 a
−→ +
c cz + d c
az + b
= = T (z).
cz + d
This way of breaking a bilinear transformation into simpler components has two purposes.
First, we can analyze general properties of these transformations by analyzing properties of the

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October 14, 2010 15:39 THM/NEIL Page-760 27410_23_ch23_p751-788