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756 CHAPTER 23 Conformal Mappings and Applications

z

w

γ
z 0 f(γ)
f(z )
0

D D *

FIGURE 23.9 A curve in a domain and its image
under f .
f(γ)
y γ v
z

w
z 0 w 0
θ ϕ
x u

FIGURE 23.10 Argument θ of the line through z and z 0 .

for z in γ , is a smooth curve through f (z 0 ) in D (Figure 23.9). Let z be another point on γ , and
∗
write w = f (z) and w 0 = f (z 0 ). Then
f (z) − f (z 0 )
w − w 0 = (z − z 0 ).
z − z 0
Now, the argument of a product of two numbers is the sum of the arguments of the numbers (to
within integer multiples of 2π). Therefore, to within 2nπ,
f (z) − f (z 0 )

arg(w − w 0 ) = arg + arg(z − z 0 ).
z − z 0
In Figure 23.10, θ is the angle between the positive real axis and the line through z and z 0
and is an argument of z − z 0 . The angle ϕ between the positive real axis and the line through w
and w 0 is an argument of w − w 0 . In the limit, as z → z 0 , the last equation implies that

ϕ = arg( f (z 0 )) + θ.
Repeat this discussion for any other smooth curve γ through z 0 with z a point on γ .Bythe
∗
∗
∗
same reasoning,

∗
∗
ϕ = arg( f (z 0 )) + θ .
But then
∗
∗
ϕ − ϕ = θ − θ .
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October 14, 2010 15:39 THM/NEIL Page-756 27410_23_ch23_p751-788