2.1 • FUNCTIONS AND LINEAR MAPPINC S 55We now show how to find the image B of a specified set A under a givenmapping u+iv = w = f (z). The set A is usually described with an equation or
inequality involving x and y. Using inverse functions, we can construct a chain
of equivalent statements leading to a description of the set B in terms of an
equation or an inequality involving u and v.
- EXAMPLE 2.7 Show that the function f (z) = iz maps the line y = x + 1
in the xy plane onto the line v = -u - 1 in the w plane.
Solution (Method 1): With A = {(x, y) : y = x + l}, we want to describe
B = f (A). We let z = x + iy E A and use Equations (2-5) and Example 2.6 to
get
u+iv = w = f(z ) E B= 1-^1 (w) = z = x+iy E A
= - iwE A
=v-iuE A
= (v , - u) E A
= - u =v+ 1
= v = - ·u-1,where = means "if and only if" {iff).
Note what this result says: u + iv = w E B = v = - u -1. The image of
A under f, therefore, is the set B = {(u, v) : v = -u - 1}.
(Method 2): We write u+iv = w = f (z) = i(x+iy) = -y+i x and note that
the transformation can be given by the equations u = - y and v = x. Because A
is described by A= {x + iy: y = x + 1}, we can substitute u = - y and v = x
into the equation y = x + 1 to obtain -u = v + 1, which we can rewrite as
v = - u - 1. H you use this method, be sure to pay careful attention to domains
and ranges.
We now look at some elementary mappings. If we let B = a + ib denote a
fixed complex constant, the transformationw = T (z) = z + B = x + a + i (y + b)
is a one-to-one mapping of the z plane onto thew plane and is called a transla-
tion. This transformation can be visualized as a rigid translation whereby the
point z is displaced through the vector B = a + ib to its new position w = T (z).
The inverse mapping is given by
z = r -^1 (w) = w - B = ·u - a + i (v - b)
and shows that T is a one-to-one mapping from the z plane onto the w plane.
The effect of a translation is depicted in Figure 2.5.