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23.1 Conformal Mappings 755

z w z L 2
C 2 Tangent to f(C ) w
2
f(L )
Tangent to C 2 2
θ θ f(C 2 ) L 1
Tangent to C 1 Tangent to f(L )
1
f(C )
1
C 1 f(C 1 )

FIGURE 23.6 f preserves angles. FIGURE 23.7 f preserves orientation.

z L 2 w

f(L )
2
L 1
f(L )
1

FIGURE 23.8 f fails to preserve orientation.

Not every function preserves angles and orientation. For example, the function f (z) = z
reverses a rotation from counterclockwise to clockwise orientation. In particular, if we think of
the counterclockwise rotation from z = 0to z = i, the image points f (0) = 0 and f (i) = i =−i
has a reversed, clockwise sense of rotation.

A function that is both orientation and angle preserving on a domain is said to be conformal
on this domain. We refer to such a function as a conformal mapping.

The next theorem provides a large number of conformal mappings, namely, differentiable
functions with nonvanishing derivatives.

THEOREM 23.1 Conformal Mappings

Let D and D be domains and let f : D → D . Suppose f is differentiable on D and f (z) = 0

∗
∗
on D. Then f is a conformal mapping.
We will sketch a geometric argument to suggest why this result is true. Let z 0 be in D
and let γ be a smooth curve in D through z 0 . Then f (γ ), which consists of all points f (z)

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October 14, 2010 15:39 THM/NEIL Page-755 27410_23_ch23_p751-788