Complex Analysis - Maths KU

100 Chapter 2 Complex Functions and Mappings
θ
r
– θ
1/r
1/z
Figure 2.39 The reciprocal mapping
z
(a) Use mappings to determine upper and lower bounds on the modulus of
f(z).
(b) Find values of z that attain your bounds in (a). In other words, find z0
and z1 such that |f(z0)| = L and |f(z1)| = M.
2.5 Reciprocal Function
In Sections 2.32.5 and 2.4 we examined some special types of complex polynomial functions
as mappings of the complex plane. Analogous to real functions, we define a complex
rational function to be a function of the form f(z) =p(z)/q (z) where both p(z) and q(z)
are complex polynomial functions. In this section, we study the most basic complex rational
function, the reciprocal function 1/z, as a mapping of the complex plane. An important
propertyof the reciprocal mapping is that it maps certain lines onto circles.
Reciprocal Function The function 1/z, whose domain is the set of
all nonzero complex numbers, is called the reciprocal function. To study
the reciprocal function as a complex mapping w =1/z, we begin byexpressing
this function in exponential notation. Given z �= 0,ifwesetz = re iθ , then
we obtain:
w = 1
z
1 1
= =
reiθ r e−iθ . (1)
From (1), we see that the modulus of w is the reciprocal of the modulus of z
and that the argument of w is the negative of the argument of z. Therefore,
the reciprocal function maps a point in the z-plane with polar coordinates
(r, θ) onto a point in the w-plane with polar coordinates (1/r, −θ). In Figure
2.39, we illustrate the relationship between z and w =1/z in a single copyof
the complex plane. As we shall see, a simple wayto visualize the reciprocal
function as a complex mapping is as a composition of inversion in the unit
circle followed by reflection across the real axis. We now proceed to define
and analyze each of these mappings.
Inversion in the Unit Circle The function
g(z) = 1
r eiθ , (2)
whose domain is the set of all nonzero complex numbers, is called inversion
in the unit circle. We will describe this mapping byconsidering separately
the images of points on the unit circle, points outside the unit circle, and
points inside the unit circle. Consider first a point z on the unit circle. Since
z =1· eiθ , it follows from (2) that g (z) = 1
1eiθ = z. Therefore, each point
on the unit circle is mapped onto itself by g. If, on the other hand, z is a
nonzero complex number that does not lie on the unit circle, then we can
write z as z = reiθ with r �= 1. When r>1 (that is, when z is outside of the