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Introduction to complex analysis

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# 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 ‘’.

## Box _unit1.2.1 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

1. 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.

2. The derivative is sometimes written as or .

3. 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

• a.the constant function

• b.the function .

• a. 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

• b. 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.

## Box _unit1.2.2 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.

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.

## Box _unit1.2.3 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.

Figure _unit1.2.1 Figure 3 Three regions on which is analytic

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.

• a.An entire function is analytic at every point of .

• b.If a function is differentiable at each point of a set, then it is analytic on that set.

• a.True.

• b.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.

## Box _unit1.2.4 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.

## Box _unit1.2.5 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.