Complex Analysis - Maths KU

80 Chapter 2 Complex Functions and Mappings
(ii) there exists an element e in G such that e ∗ a = a ∗ e = a for all a in
G, and
(iii) for every element a in G there exists an element b in G such that
a ∗ b = b ∗ a = e. (The element b is called the inverse of a in G and is denoted
by a −1 .)
Let Isom+(E) denote the set of all complex functions of the form f(z) =az + b
where |a| = 1. In the remaining part of this project you are asked to demonstrate
that Isom+(E) is a group with composition of functions as the binary
operation. This group is called the group of orientation-preserving isometriesof
the Euclidean plane.
(b) Prove that composition of functions is a binary operation on Isom+(E).
That is, prove that if f and g are functions in Isom+(E), then the function
f ◦ g defined by f ◦ g(z) =f (g(z)) is an element in Isom+(E).
(c) Prove that the set Isom+(E) with composition satisfies property (i) ofa
group.
(d) Prove that the set Isom+(E) with composition satisfies property (ii) ofa
group. That is, show that there exists a function e in Isom+(E) such that
e ◦ f = f ◦ e = f for all functions f in Isom+(E).
(e) Prove that the set Isom+(E) with composition satisfies property (iii) ofa
group.
2.4 Special Power Functions
A complex polynomial 2.4 function is a function of the form p(z) =anzn +an−1zn−1 +...+
a1z + a0 where n is a positive integer and an, an−1, ... , a1, a0 are complex constants. In
general, a complex polynomial mapping can be quite complicated, but in many special cases
the action of the mapping is easilyunderstood. For instance, the complex linear functions
studied in Section 2.3 are complex polynomials of degree n =1.
In this section we studycomplex polynomials of the form f(z) =zn , n ≥ 2. Unlike
the linear mappings studied in the previous section, the mappings w = zn , n ≥ 2, do not
preserve the basic shape of everyfigure in the complex plane. Associated to the function
zn , n ≥ 2, we also have the principal nth root function z1/n . The principal nth root
functions are inverse functions of the functions zn defined on a sufficientlyrestricted domain.
Consequently, complex mappings associated to zn and z1/n are closelyrelated.
Power Functions Recall that a real function of the form f(x) =xa ,
where a is a real constant, is called a power function. We form a complex
power function byallowing the input or the exponent a to be a complex
number. In other words, a complex power function is a function of the
form f(z) =zα where α is a complex constant. If α is an integer, then the
power function zα can be evaluated using the algebraic operations on complex
numbers from Chapter 1. For example, z2 = z · z and z−3 1
= . We
z · z · z
can also use the formulas for taking roots of complex numbers from Section
1.4 to define power functions with fractional exponents of the form 1/n. For