Complex Analysis - Maths KU

50 Chapter 2 Complex Functions and Mappings
Notation used throughout this text.
2.1 Complex Functions
One of the most 2.1 important concepts in mathematics is that of a function. You mayrecall
from previous courses that a function is a certain kind of correspondence between two sets;
more specifically:
A function f from a set A to a set B is a rule of correspondence that assigns
to each element in A one and only one element in B.
We often think of a function as a rule or a machine that accepts inputs from the set A and
returns outputs in the set B. In elementarycalculus we studied functions whose inputs and
outputs were real numbers. Such functions are called real-valued functions of a real
variable. In this section we begin our studyof functions whose inputs and outputs are
complex numbers. Naturally, we call these functions complex functions of a complex
variable, orcomplex functions for short. As we will see, manyinteresting and useful
complex functions are simplygeneralizations of well-known functions from calculus.
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Function Suppose that f is a function from the set A to the set B. If
f assigns to the element a in A the element b in B, then we saythat b is the
image of a under f, or the value of f at a, and we write b = f(a). The
set A—the set of inputs—is called the domain of f and the set of images in
B—the set of outputs—is called the range of f. We denote the domain and
range of a function f by Dom(f) and Range(f), respectively. As an example,
consider the “squaring” function f(x) =x 2 defined for the real variable x.
Since anyreal number can be squared, the domain of f is the set R of all
real numbers. That is, Dom(f) =A = R. The range of f consists of all real
numbers x 2 where x is a real number. Of course, x 2 ≥ 0 for all real x, and
it is easyto see from the graph of f that the range of f is the set of all
nonnegative real numbers. Thus, Range(f) is the interval [0, ∞). The range
of f need not be the same as the set B. For instance, because the interval
[0, ∞) is a subset of both R and the set C of all complex numbers, f can be
viewed as a function from A = R to B = R or f can be viewed as a function
from A = R to B = C. In both cases, the range of f is contained in but not
equal to the set B.
As the following definition indicates, a complex function is a function
whose inputs and outputs are complex numbers.
Definition 2.1 Complex Function
A complex function is a function f whose domain and range are subsets
of the set C of complex numbers.
A complex function is also called a complex-valued function of a
complex variable. For the most part we will use the usual symbols f, g,
and h to denote complex functions. In addition, inputs to a complex function
f will typicallybe denoted bythe variable z and outputs bythe variable
w = f(z). When referring to a complex function we will use three notations