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23.1 Conformal Mappings 757

Since θ − θ is the angle between the tangents to γ and γ at z 0 and ϕ − ϕ is the angle between
∗
∗
∗
the tangents to f (γ ) and f (γ ) at f (z 0 ), this shows that f preserves angles. The assumption that
∗
f (z 0 ) = 0 is required to take the argument of f (z 0 ).

This equation also shows that f preserves orientation, since the sense of rotation from γ to
γ is the same as that from f (γ ) to f (γ ), otherwise we would have
∗
∗
∗
∗
ϕ − ϕ =−(θ − θ ).
This shows that f is conformal.
A mapping f : D → D is one-to-one, often written 1 − 1, if two different numbers in D
∗
cannot be mapped to the same number in D .
∗

This means that f (z 1 ) = f (z 2 ) if z 1 and z 2 are different points of D. Thus, for example,
w = sin(z) is not a one-to-one mapping of the complex plane to the complex plane, because
f (0) = f (2π) = 0.
∗
∗
We say that f is onto D if every number in D is the image of some number in D under the
mapping. This means that if w is in D there must be some z in D such that f (z) = w.
∗
If f is a one-to-one mapping of a domain D onto a domain D , then there is a unique pairing
∗
∗
of each z in D with exactly one w in D , and conversely. This enables us to define the inverse
∗
mapping f −1 : D → D by setting
f −1 (w) = z exactly when f (z) = w.
It is possible to show that f −1 is conformal if f is.
∗
If f : D → D and g : D → D ∗∗ are both conformal, then their composition g ◦ f : D →
∗
D ∗∗ is also conformal, since angles and orientation are preserved at each stage of the mapping
(Figure 23.11).

f
D *
D

g f g
°

D **

FIGURE 23.11 A composition of conformal map-
pings is conformal.

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October 14, 2010 15:39 THM/NEIL Page-757 27410_23_ch23_p751-788