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758 CHAPTER 23 Conformal Mappings and Applications

Sometimes we want to construct a conformal mapping between two sets—often between
two domains D and D . We will see that this is one approach to solving Dirichlet problems.
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Depending on D and D , constructing a conformal mapping f : D → D may be very difficult.
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There is a special class of relatively simple mappings that can sometimes be used.

A mapping T is called a bilinear transformation,or bilinear mapping if it has the form
az + b
w = T (z) =
cz + d
for given constants a,b,c, and d with ad −bc =0. This condition insures that we can solve
for z and invert the mapping
dw − b
−1
z = T (w) = ,
−cw + a
which is again a bilinear transformation. Bilinear transformations are also known as linear
fractional transformations or Möbius transformations, although these terms sometimes
carry slight variations in the definition.

Because
ad − bc

T (z) =
(cz + d) 2
is not zero, a bilinear transformation and its inverse are conformal.
We will look at some special kinds of bilinear transformations, with a view toward dissecting
general bilinear transformations into simple components.

EXAMPLE 23.4
Let w = T (z) = z + b, with b constant. This mapping is called a translation, because T shifts z
horizontally by Re(z) and vertically by Im(z).
As an example, let T (z) = 2 − i. T takes z and moves it two units to the right and one unit
down (Figure 23.12). For example, T maps
0 → 2 − i, 1 → 3 − i, i → 2, 4 + 3i → 6 + 2i.

y

z

w = z + 2 − i

x

FIGURE 23.12 A translation.

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October 14, 2010 15:39 THM/NEIL Page-758 27410_23_ch23_p751-788