Chapter 3  Analytic and Harmonic Functions

Overview

        Does the notion of a derivative of a complex function make sense?  If so, how should it be defined and what does it represent?  
        
These and similar questions are the focus of this chapter.  As you might guess, complex derivatives have a meaningful definition, and

many of the standard derivative theorems from from calculus (such as the product rule and chain rule) carry over for complex functions.  

There are also some interesting applications.  But not everything is symmetric.  You will learn in this chapter that the mean value theorem

for derivatives does not extend to complex functions.  In later chapters you will see that differentiable complex functions are, in some sense,

much more "differentiable" than differentiable real functions.

Section 3.1  Differentiable and Analytic Functions

        Using our imagination, we take our lead from elementary calculus and define the derivative of    at  ,  written  ,  by  

(3-1)               ,

provided that the limit exists.  If it does, we say that the function    is differentiable at  .  If we write  ,  

then we can express Equation (3-1) in the form

(3-2)               .  
  
    If we let    and  ,  then we can use the Leibniz's notation for the derivative:  

(3-3)               .  

Explore the Derivative.

Example 3.1.  Use the limit definition to find the derivative of   .  

Solution.  Using Equation (3-1), we have  

                      

We can drop the subscript on    to obtain      as a general formula.

Alternative Solution.  Using Equation (3-2), we have  

                      

We can drop the subscript on    to obtain      as a general formula.

Explore Solution 3.1.

        Pay careful attention to the complex value    in Equation (3-3);  the value of the limit must be independent of

the manner in which  .   If we can find two curves that end at    along which    approaches two distinct values,

then    does not have a limit as    and   does not have a derivative at  .  

The same observation applies to the limits in Equations (3-2) and (3-1).

Example 3.2.  Show that the function      is nowhere differentiable.

Solution.  We choose two approaches to the point    and compute limits of the two difference quotients.  

We shall use formulas similar to (3-1), for calculating the directional derivatives along horizontal and vertical lines.

First, we approach    along a line parallel to the -axis by forcing to be of the form  .   

                        

Next, we approach along a line parallel to the -axis by forcing to be of the form  .   

                       

        The limits along the two paths are different, so there is no possible value for the right side of Equation (3-1).  

Therefore   is not differentiable at the point , and since was arbitrary,     is nowhere differentiable.

Explore Solution 3.2.

Remark 3.1.  In Section 2.3 we showed that      is continuous for all  .  Thus, we have a simple example of a

function that is continuous everywhere but differentiable nowhere.  Such functions are hard to construct in real variables.  

In some sense, the complex case has made pathological constructions simpler!

        We are seldom interested in studying functions that aren't differentiable, or even differentiable at only a single point.  

Complex functions that have a derivative at all points in a neighborhood of    deserve further study.  In Section 7.2,

we will prove that, if the complex function    can be represented by a Taylor series at  ,  then it must be

differentiable in some neighborhood of  .  Functions that are differentiable in neighborhoods of points are pillars

of the complex analysis edifice; we give them a special name, as indicated in the following definition.

Definition 3.1 (Analytic Function).  The complex function    is analytic at the point    provided there

is some    such that    exists for all  .   In other words,    must be differentiable

not only at  ,  but also at all points in some -neighborhood of  .

        If    is analytic at each point in the region  ,  then we say that    is an analytic function on  .  

Again, we have a special term if    is analytic on the whole complex plane.

Definition 3.2 (Entire Function).  If    is analytic on the whole complex plane then    is said to be an entire function.

        Points of non-analyticity for a function are called singular points. They are important for applications in physics and engineering.  

        Our definition of the derivative for complex functions is formally the same as for real functions and is the natural extension from  

real variables to complex variables.  The basic differentiation formulas are identical to those for real functions, and we obtain the same

rules for differentiating powers, sums, products, quotients, and compositions of functions.  We can easily establish the proof of the

differentiation formulas by using the limit theorems.

The Rules for Differentiation.

        Suppose that f(z) and g(z) are differentiable.  From Equation (3-2) and the technique exhibited in the solution to Example 3.1

we can establish the following rules, which are virtually identical to those for real-valued functions.  


(3-4)               ,  

(3-5)               ,  

(3-6)               ,  

(3-7)               ,  

(3-8)               ,  

(3-9)               ,  

(3-10)              .  

Important particular cases of Equations (3-9) and (3-10), respectively, are  

(3-11)              ,  

(3-12)              .  

Exploration for the Rules for Differentiation.

Example 3.3.  Use Formula (3-12) to calculate  .      

Hint.  Use    ,    ,   and   .  

Solution.  An easy computation yields  

                      

Explore Solution 3.3.

        The proofs of the rules given in Equations through (3-10) depend on the validity of extending theorems for real functions to their

complex companions.  Equation (3-8), for example, relies on Theorem 3.1.

Theorem 3.1.  If    is differentiable at    then    is continuous at  .  

Proof.  From Equation (3-1), we obtain  

                    .  

Using the multiplicative property of limits given in Theorem 2.3 in Section 2.3, we get

                      

This result implies that    ,  which in turn implies that

                     .  

Therefore,    is continuous at .

Proof.

The Derivative of  

        We can establish Equation (3-8)  ,  from Theorem 3.1.  

Letting      and using Definition 3.1, we write   

                    .  

If we subtract and add the term    in the numerator, we get

                       

Using the definition of the derivative given by Equation (3-1) and the continuity of  ,  we obtain  

                    ,  

which is what we wanted to establish.  

        We leave the proofs of the other differentiation rules as exercises.

        The rule for differentiating polynomials carries over to the complex case as well.  

If we let    be a polynomial of degree , so that    

                    ,  

then mathematical induction, along with Equations (3-5) and (3-7), gives  

                    .  

Again, we leave the details of this proof for the reader to finish, as an exercise.

        We shall use the differentiation rules as aids in determining when functions are analytic.  For example, Equation (3-9) tells us

that if    are polynomials, then their quotient    is analytic at all points where  .  This condition

implies that the function      is analytic for all  .  

        The square root function is more complicated.   If   ,   then    is analytic at

all points except    (because is undefined)  and at points that lie along the negative -axis.  Recall from Exercise 17,

in Section 2.3, that the argument function  is not continuous along the negative -axis.  Therefore the function  ,  

is not continuous at points that lie along the negative -axis.

        We close this section with a complex extension of a famous theorem, which is attributed to Guillaume de l'Hôpital (1661-1704),

the proof of will be given in Section 7.5.  

Theorem 3.2 (L'Hôpital's Rule).  Assume that    and    are both analytic at  .  

If   ,   ,   and   ,  then  

                    .  

Proof.

Exploration for L'Hôpital's Rule.

Extra Example 1.  Use L'Hôpital's rule to find   .  

Explore Extra Example 1.

Optional Material for the Internet

Real Concepts in Complex Analysis.

        Many of the calculus concepts about derivatives are easy to extended to complex functions.  For example, in calculus

we learned that the derivative is the limit of the difference quotients    as    goes to zero.  We can compare

our calculus experience with some new and interesting graphs in the complex plane.

Graphical explorations of difference quotients.

Extra Example 2.  Consider the real function  

                    ,  
                    
which is differentiable, and it's derivative is the limit of the real difference quotients  .

                    

        We can illustrate convergence of the real difference quotients    by comparing graphs for decreasing

values of  .  For illustration purposes we plot the real graphs      for   .  

                                                            The graph of   .

                                   Figure E.E.3.  The graphs of      for   .  

                                   where        and the graph of    .  

        By looking at the above graphs we should get a good feeling about visualizing limits of functions over an interval.  In particular,

we hope that this gives you a good feeling about the the formula    ,    where we have used

the function        in this illustration to get    .  

        The real function can be extended into the complex plane by replacing the real variable with the complex variable  .

The same algebraic computations are involved in finding the limit of the complex difference quotients.  

Extra Example 3.  Consider the complex function  

                    ,  
                    
which is differentiable, and it's derivative is the limit of the complex difference quotients  .

                    

        We can illustrate convergence of the complex difference quotients      by comparing graphs for decreasing values

of  .  For illustration purposes we plot the graphs      for   .

We cannot draw a graph of -dimensional space into -dimensional space, it is necessary to choose a domain in the -plane for our graphs.

                                                                 The domain in the -plane for the following graphs.  

                                                                 The graph of   .

                              Figure E.E.4.  The unit square in the -plane, and it's images under the mappings

                                  for    ,  
                                        
                              where        and the graph of    .

        By looking at the above graphs we should get a good feeling about visualizing limits of functions in the complex plane.  

In particular, we hope that this gives you a good feeling about the the formula   ,   where we

have used the function       in this illustration to get    .  

Remark.  The final resting place of the points    are   

                    ,    ,   

                    ,   and   .  

Graphical Explorations of Polynomial Approximations.

        Many concepts from calculus will be extended to complex functions, including the approximation of functions.  Derivatives will  

play an important role, just as they did in the calculus of real functions.  Let us give a preview of some things we will be studying.

The following three polynomial approximations are usually discussed in calculus.

                                           is an approximation to   .  

                                           is an approximation to   .  

                                           is an approximation to   .  

        In the above graphs, is easy to visualize the real functions and their polynomial approximations    

                                        ,  

                                        ,  

                                        .  
                              
We assume that the reader is familiar with the details for constructing these approximations, or can easily find them.

        However, when we extend these real functions to complex functions, we must select an appropriate

domain for each function in the -plane in order to construct it's graph.  The following complex function

examples give illustrations similar to the above real approximations, but extended into the complex plane.  

Extra Example 4.  Given  ,  from calculus we know that  ,   .  

The Maclaurin polynomial of degree    is   .

Hence, the mapping   ,   has the "linear approximation"   .

                                                       The domain in the -plane for the following graphs.  

                    Figure E.E.5.  The domain      is a square in the -plane,  

                    and it's images under the mappings        and    .  

                    In the last two graphs, one can visualize the complex function approximation

                    .  

Remark 1.  This is a trivial example of  a "linear transformation" that was studied in Section 2.1.  

Also,    is a "linear approximation" to  .

Remark 2.  Complex Taylor polynomials and approximations will be introduced in Section 7.2.  

The function    is the familiar Maclaurin polynomial approximation of degree  .

Remark 3.  Analytic functions that satisfy    are conformal mappings and will be studied in Section 10.1.  

Extra Example 5.  Given  ,  from calculus we know that    ,    ,    .  

The Maclaurin polynomial of degree    is   .

Hence, the mapping   ,   has the "quadratic approximation"   .

                                                       The domain in the -plane for the following graphs.  

                    Figure E.E.6.  The domain      is a rectangle in the -plane,  

                    and it's images under the mappings        and    .

                    In the last two graphs, one can visualize the complex function approximation

                    .  

Remark 4.  The mapping    is similar to the mapping    that was studied in Section 2.2.  

Also,    is a "quadratic approximation" to  .

Remark 5.  Complex Taylor polynomials and approximations will be introduced in Section 7.2.  

The function    is the familiar Maclaurin polynomial approximation of degree  .

Remark 6.  Analytic functions that satisfy    are conformal mappings and will be studied in Section 10.1.  

We will see that the mapping      is not conformal at the origin.

Extra Example 6.  Given  ,  from calculus we know that the first few derivatives are

                    ,    ,     ,    .  

The Maclaurin polynomial of degree    is   .

Hence, the mapping   ,   has the "cubic approximation"   .

                                                       The domain in the -plane for the following graphs.  

                    Figure E.E.7.  The domain      is a square in the -plane,  

                    and it's images under the mappings        and    .  

                    In the last two graphs, one can visualize the complex function approximation

                    .  

Remark 7.  Complex Taylor polynomials and approximations will be introduced in Section 7.2.  

The function    is the familiar Maclaurin polynomial approximation of degree  .  

Remark 8.  Analytic functions that satisfy    are conformal mappings and will be studied in Section 10.1.  

Exercises for Section 3.1.  Differentiable and Analytic Functions

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell