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752 CHAPTER 23 Conformal Mappings and Applications

z w

f

K
S

FIGURE 23.1 f (z) as a mapping.

We will examine this mapping perspective for two familiar functions.

EXAMPLE 23.2
z
Let f (z) = e . Then
z
x
x
w = u + iv = e = e x+iy = e cos(y) + ie sin(y),
so
x
x
u = e cos(y) and v = e sin(y).
Consider a vertical line x =a in the z-plane. The image of this line under the exponential mapping
consists of points u + iv with
a
a
u = e cos(y) and v = e sin(y).
Now
2a
2
2
u + v = e ,
a
so this vertical line x = a maps to the circle of radius e about the origin in the w-plane
a iy
(Figure 23.2). As the point z = a + iy moves along the line, the image point e a+iy = e e moves
around this circle, making one complete circuit every time y varies over an interval of length
2π. We may think of this vertical line as made up of infinitely many intervals of length 2π
strung together, and the exponential function maps each of these segments once around the image
circle.
The image of a point z = x + ib on a horizontal line y = b is the point u + iv with
x
x
u = e sin(b) and v = e cos(b).

z w
w = e z v
y

x u

a
w = e
Re(z) = a
FIGURE 23.2 Image of a vertical line under the map-
z
ping f (z) = e in Example 23.2.

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