1.1 Defining differentiable functions
As with limits and continuity, the way in which the derivative of a complex function is defined is similar to the real case. Thus a complex function is said to have a derivative at a point if the difference quotient, defined by

tends to a limit as tends to
. Equivalently, it is sometimes more convenient to replace
by
, and examine the corresponding limit as
tends to 0. The difference quotient then has the form

where is a complex number. The equivalence of these two limits can be justified by noting that if
, then ‘
’ is equivalent to ‘
’.
Definitions
Let be a complex function whose domain contains the point
. Then the derivative of
at
is

provided that this limit exists. If it does exist, then is differentiable at
. If
is differentiable at every point of a set
, then
is differentiable on
. A function is differentiable if it is differentiable on its domain.
The derivative of at
is denoted by
, and the function

is called the derivative of . The domain of
is the set of all complex numbers at which
is differentiable.
The function is sometimes called the derived function of
.
Remarks
The existence of the limit
implicitly requires the domain of
to contain
as one of its limit points. This always holds if the domain of
is a region.
The derivative
is sometimes written as
or
.
Some other texts use the phrase complex derivative in place of derivative to draw a distinction with the standard real derivative of a function
(which we will not need).
In certain cases it is easy to find the derivative of a function directly from the definition above.
Example 1
Use the definition of derivative to find the derivative of the function .
Solution
The domain of is the whole of
, so let
be an arbitrary point of
. Then

Now is a basic continuous function, continuous at
, so we see that
.
Since is an arbitrary complex number, the derivative of
is the function
. Its domain is the whole of
.
Notice the way in which the troublesome term cancels from the numerator and the denominator in the calculation of
in the preceding example. This often happens when you calculate derivatives directly from the definition.
Exercise 1
Use the definition of derivative to find the derivative of
the constant function
the function
.
Answer
is defined on the whole of
, so let
. Then
Since
is an arbitrary complex number,
is differentiable on the whole of
, and its derivative is the zero function
is defined on the whole of
, so let
. Then
Since is an arbitrary complex number,
is differentiable on the whole of
, and its derivative is the constant function

Example 1 and Exercise 1 show that the functions ,
and
are differentiable on the whole of
. Functions that have this property are given a special name.
Definition
A function is entire if it is differentiable on the whole of .
Not all functions are entire; indeed, many interesting aspects of complex analysis arise from functions that fail to be differentiable at various points of .
Exercise 2
Use the definition of derivative to find the derivative of the function . Explain why
is not entire.
Answer
The domain of is the region
. Since
cannot exist unless
is defined at
, we confine our attention to
. Then

Now is a basic continuous function with domain
, so we see that

Since is an arbitrary non-zero complex number, the derivative of
is

The function is not entire since its domain is not
.
Although the function is not entire, it is differentiable on the whole of its domain
. This domain is a region because it is obtained by removing the point 0 from
. (The removal of a point from a region leaves a region.) As the course progresses, you will discover that regions provide an excellent setting for analysing the properties of differentiable functions. We therefore make the following definitions.
Definitions
A function that is differentiable on a region is said to be analytic on
. If the domain of a function
is a region, and if
is differentiable on its domain, then
is said to be analytic. A function is analytic at a point
if it is differentiable on a region containing
.
It follows immediately from the definition that if a function is analytic on a region , then it is automatically analytic at each point of
.
Notice that a function can have a derivative at a point without being analytic at the point. For example, in the next section we will ask you to show that the function has a derivative at 0, but at no other point. This means that there is no region on which
is differentiable, and hence no point at which
is analytic.
By contrast, is analytic at every point of its domain. It is an analytic function, and it is analytic on any region that does not contain 0. Three such regions are illustrated in Figure 3.


An appropriate choice of region can often simplify the analysis of complex functions.
Exercise 3
Classify each of the following statements as True or False.
An entire function is analytic at every point of
.
If a function is differentiable at each point of a set, then it is analytic on that set.
Answer
True.
False. (The set must be a region.)
There is a close connection between differentiation and continuity. The function , for example, is not only differentiable, but also continuous on its domain. This is no accident for, as in real analysis, differentiability implies continuity.
Theorem 1
Let be a complex function that is differentiable at
. Then
is continuous at
.
Proof
Let be differentiable at
; then

To prove that is continuous at
, we will show that
as
. We do this by proving the equivalent result that
as
.
By the Product Rule for limits of functions, we have

Hence as
, so
is continuous at
.
In fact, differentiability implies more than continuity. Continuity asserts that for all close to
,
is close to
. For differentiable functions, this ‘closeness’ has the ‘linear’ form described in the following theorem.
Theorem 2 Linear Approximation Theorem
Let be a complex function that is differentiable at
. Then
can be approximated near
by a linear polynomial. More precisely,

where is an ‘error function’ satisfying
as
.
Informally speaking, the statement ‘ as
’ means that ‘
tends to zero faster than
does’.
Proof
We have to show that the function defined by

satisfies as
.
Dividing by
and letting
tend to
, we obtain

as required.
Theorem 1 and Theorem 2 are often used to investigate the properties of differentiable functions. An illustration of this occurs in the next subsection, where Theorem 1 is used in a proof of the Combination Rules for differentiation. Later in this section we use Theorem 2 to give a geometric interpretation of complex differentiation.
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