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754 CHAPTER 23 Conformal Mappings and Applications

z
w
y
v

π 2 π 2 x
−1 1 u

FIGURE 23.5 Image of a strip under f (z) = sin(z) in
Example 23.3.

Now look at boundary points of S. The boundary of S consists of the segment −π/2 ≤ x ≤
π/2 on the real axis together with the half-lines x =−π/2 and x = π/2 with y ≥ 0. Imagine
a point z moving counterclockwise (positive orientation) around the boundary of S, and keep a
record of how the image point w moves. Follow z and its image in Figure 23.5.
To start, put z on the left vertical boundary of S, which is the half-line x =−π/2 with y ≥ 0.
This z maps to

w = u + iv =−cosh(y).
Furthermore, as z moves down this half-line toward the real axis so y is decreasing toward zero,
w =−cosh(y) moves from left to right on the negative real axis and approaches −1 from the left.
Now z has reached −π/2 and turns to move from −π/2to π/2 along the bottom boundary
segment of S. On this segment, z = x and y = 0, so the image point is w = sin(x), which moves
from −1to1as z moves toward π/2.
When z reaches π/2, it turns and moves up the right side of S, which is the half-line x =
π/2, y ≥ 0. The image point is w = cosh(y), which moves from 1 to the right out the real axis in
the w-plane as y increases.
Therefore, f maps the boundary of S to the boundary of the upper half-plane, which is the
real axis in the w-plane. If we imagine walking around the boundary of S in a positive sense
with S over our left shoulder, the image point moves in a positive sense over the boundary of
the image of S, which is the upper half-plane (which is over our left shoulder in that plane if we
walk left to right along the real axis).

We will find two properties enjoyed by some mappings to be particularly important.

1. f preserves angles if it satisfies the following requirement. If L 1 and L 2 are smooth
curves in S intersecting at a point z 0 in S and θ is the angle between their tangents at
this point, the images f (L 1 ) and f (L 2 ) in the w-plane have the same angle θ between
their tangents at f (z 0 ). This is indicated in Figure 23.6.
2. f preserves orientation if the following is true. In the scenario of property (1), if
the sense of orientation from L 1 to L 2 at any point z 0 is counterclockwise, the sense
of orientation from f (L 1 ) to f (L 2 ) in the w-plane also must be counterclockwise.
Figure 23.7 shows the idea of an orientation preserving map, while Figure 23.8 suggests
an orientation reversing map.

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October 14, 2010 15:39 THM/NEIL Page-754 27410_23_ch23_p751-788