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)            .  

Extra Example 1.  Use the limit definition to find the derivative  .  

Explore Extra Example 1.

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.

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 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 difference quotients.  First, we approach along a line parallel to the x axis by forcing z to be of the form .   

                

Next, we approach along a line parallel to the y axis by forcing z 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 z.  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 seldom are interested in studying functions that aren't differentiable, or are differentiable at only a single point.  Complex functions that have a derivative at all points in a neighborhood of deserve further study.  In Chapter 7 we demonstrate that, if the complex function f(z) 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 f(z) is analytic at each point in the region R, then we say that f(z) is analytic on R.  Again, we have a special term if f(z) is analytic on the whole complex plane.

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

    Points of nonanalyticity for a function are called singular points. They are important for certain 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.  

  

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

  

Exploration for the Rules for Differentiation.

Example 3.3.  If we use Equation (3-12) with  ,  and  ,  then we get  

              

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 f(z) is differentiable at    then f(z) is continuous at  .  

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

            .  

Using the multiplicative property of limits given by Formula (2-19), we get

              

This result implies that  ,  which is equivalent to showing that f(z) is continuous at .

Proof.

    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 rules as exercises.

    The rule for differentiating polynomials carries over to the complex case as well.  If we let P(z) be a polynomial of degree n, so   

            ,  

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

            .  

Again, we leave the proof of this result as an exercise.

    We can 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 x axis.  The argument function, and therefore the function f(z) itself, are not continuous at points that lie along the negative x axis.

    We close this section with a complex extension of a famous theorem, which is attributed to Guillaume François Antoine Marquis de L'Hôpital (1661-1704).  The proof of will be given in Chapter 7.  

Theorem 3.2, (L'Hôpital's Rule).  Assume that f(z) and g(z) are both analytic at .  If , , and   then  

            .  

Proof.

Exploration for L'Hospital's Rule.

Extra Example 2.  Use L'Hopital's rule to find  .  

Explore Extra Example 2.

Exercises for Section 3.1.  Differentiable and Analytic Functions