Page 779 - Advanced_Engineering_Mathematics o'neil

23.1 Conformal Mappings 759

w
v
z w
z 4i y v w = T(z)
y
2 + 2i z
1 + i
i z
−2 + 2i 2 + 2i
θ θ + π/4
x u
1 x u

FIGURE 23.14 Action of T(z) = (2 + 2i)zon arbi-
trary z in Example 23.5.
FIGURE 23.13 Effect of the mapping T (z)=(2+2i)z
on specific points of Example 23.5.

EXAMPLE 23.5
Let w = T (z) = az with a a nonzero constant. This mapping is called a rotation/magnification.
To see why this name applies, write
|w|=|a||z|.
If |a|>1, then T (z) is further from the origin than z is for z =0. If |a|<1, then T moves z closer
to the origin. Thus the term magnification.
But T does more than this. If we write the polar forms z =re and a = Ae , then
iα
iθ
T (z) = are i(θ+α)
adding the constant angle α to the argument of z. This corresponds to a rotation by α radians—
counterclockwise if α> 0 and counterclockwise if α< 0.
The total effect of this transformation is therefore a scaling and a rotation. We can see this
effect in the mapping

T (z) = (2 + 2i)z.
This will map, for example,
i →−2 + 2i, 1 → 2 + 2i, 1 + i →−4i,
as shown in Figure 23.13. Figure 23.14 shows the action of T on an arbitrary nonzero z, multi-
√
plying the magnitude of z by |2 + 2i|= 8 and adding arg(2 + 2i) = π/4 to the argument of z
for a counterclockwise (positive) rotation through π/4 radians.

If |a|= 1, T (z) = az is called a pure rotation. In this case, there is no magnification effect,
just a rotation through an argument of a.

EXAMPLE 23.6
Let
1
T (z) = for z = 0.
z
This mapping is called an inversion.If z = 0, then
1
|w|=
|z|

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October 14, 2010 15:39 THM/NEIL Page-759 27410_23_ch23_p751-788