Complex Analysis - Maths KU

104 Chapter 2 Complex Functions and Mappings
are near the ideal point ∞ correspond to points with extremelylarge modulus
in the complex plane.
We use this correspondence to extend the reciprocal function to a function
whose domain and range are the extended complex plane. Since (1) already
defines the reciprocal function for all points z �= 0 or ∞ in the extended
complex plane, we extend this function byspecifying the images of 0 and ∞.
A natural wayto determine the image of these points is to consider the images
of points nearby. Observe that if z = reiθ is a point that is close to 0, then
r is a small positive real number. It follows that w = 1 1
=
z r e−iθ is a point
whose modulus 1/r is large. That is, in the extended complex plane, if z is
a point that is near 0, then w =1/z is a point that is near the ideal point
∞. It is therefore reasonable to define the reciprocal function f(z) =1/z on
the extended complex plane so that f(0) = ∞. In a similar manner, we note
that if z is a point that is near ∞ in the extended complex plane, then f(z)
is a point that is near 0. Thus, it is also reasonable to define the reciprocal
function on the extended complex plane so that f(∞) =0.
Definition 2.7 The Reciprocal Function on the Extended
Complex Plane
The reciprocal function on the extended complex plane is the
function defined by:
⎧
⎪⎨
1/z, if z �= 0or∞
f(z) = ∞, if z =0
⎪⎩ 0, if z = ∞.
Rather than introduce new notation, we shall let the notation 1/z represent
both the reciprocal function and the reciprocal function on the extended
complex plane. Whenever the ideal point ∞ is mentioned, you should assume
that 1/z represents the reciprocal function defined on the extended complex
plane.
EXAMPLE 3 Image of a Line under w =1/z
Find the image of the vertical line x = 1 under the reciprocal function on the
extended complex plane.
Solution We begin bynoting that since the line x = 1 is an unbounded
set in the complex plane, it follows that the ideal point ∞ is on the line in
the extended complex plane. In Example 2 we found that the image of the
points z �= ∞ on the line x = 1 is the circle � �w − 1
�
�
2 = 1
2 excluding the point
w = 0. Thus, we need onlyfind the image of the ideal point to determine
the image of the line under the reciprocal function on the extended complex
plane. From Definition 2.7 we have that f(∞) = 0, and so w = 0 is the image