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23.1 Conformal Mappings 753

z w
y
v
Im(z) = b

w = e z
b
x u

FIGURE 23.3 Image of a horizontal line under
z
f (z) = e in Example 23.2.
z w
y θ = b
v
y = d

y = c θ = a
x u

x = b
x = a w = e b
w = e a

z
FIGURE 23.4 Image of a rectangle under f (z) = e for Exam-
ple 23.2.
x x
Because e >0 for all real x,as x +ib varies over the horizontal line, the image point e sin(b)+
x
ie cos(b) moves along a half-line from the origin to infinity, making an angle b radians with the
positive real axis (Figure 23.3). In polar coordinates, this is the half-line θ = b.
We can put these results together to find the image of any rectangle in the z-plane having
sides parallel to the axes. Let the rectangle have sides on the lines x = a, x = b, y = c, and y = d
(Figure 23.4). These lines map, respectively, to the circles
2
2
2
2
u + v = e 2a and u + v = e 2b
and the half-lines
θ = c and θ = d.
The wedge in the w-plane in Figure 23.4 is the image of this rectangle under the
exponential map.

EXAMPLE 23.3
We will determine the image, under the mapping w = f (z) = sin(z), of the strip S consisting of
all z with −π/2 ≤ Re(z) ≤ π/2 and Im(z) ≥ 0. Write

w = u + iv = sin(x)cosh(y) + i cos(x)sinh(y).
If z = x + iy is interior to S, then y > 0sosinh(y)> 0. Furthermore, since −π/2 < x <π/2for
z interior to S, cos(x)> 0. Therefore, the image point has positive imaginary part and lies in the
upper half-plane in the w-plane, so f maps the interior of S to the upper half-plane.

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