Chapter 7  Taylor and Laurent Series

Overview

    Throughout this book we have compared and contrasted properties of complex functions with functions whose domain and range lie entirely within the real numbers.  There are many similarities, such as the standard differentiation formulas.  However, there are also some surprises, and in this chapter you will encounter one of the hallmarks that distinguishes complex functions from their real counterparts:  It is possible for a function defined on the real numbers to be differentiable everywhere and yet not be expressible as a power series (see Exercise 20, Section 7.2).  For a complex function, however, things are much simpler!  You will soon learn that if a complex function is analytic in the disk , its Taylor series about converges to the function at every point in this disk.  Thus, analytic functions are locally nothing more than glorified polynomials.  

7.1  Uniform Convergence

    Complex functions are the key to unlocking many of the mysteries encountered when power series are first introduced in a calculus course.  We begin by discussing an important property associated with power series-uniform convergence.

    Recall that, for a function f(z) defined on a set T, the sequence of functions    converges to the function f(z) at the point    provided that   .  Thus, for the particular point  ,  we know that for each  ,  there exists a positive integer    (which depends on both   )  such that

(7-1)            if  ,  then  .  
    
If is the    partial sum of the series  ,  Statement (7-1) becomes  

            if  ,  then  .  

    For a given value of  ,  the integer    needed to satisfy Statement (7-1) often depends on our choice of .  This is not the case if the sequence converges uniformly.  For a uniformly convergent sequence, it is possible to find an integer    (depending only on ) that guarantees Statement  (7-1)  no matter what value for    we pick.  In other words, if n is large enough, the function is uniformly close to the function f(z) for all  .  Formally, we have the following definition.

Definition 7.1 (Uniform Convergence),.  The sequence converges uniformly to f(z) on the set  T  if for every  ,  there exists a positive integer    (depending only on )  such that

(7-2)            if  ,  then    for all   .

If    is the   partial sum of the series  ,  we say that the series    converges uniformly to  f(z)  on the set  T.

Example 7.1.  The sequence converges uniformly to the function    on the entire complex plane because for any , statement (7-2) is satisfied for all z for , where is any integer greater than .  We leave the details of showing this result as an exercise.

    A good example of a sequence of functions that does not converge uniformly is the sequence of partial sums comprising the geometric series.  Recall that the geometric series has converging to for all .  Because the real numbers are a subset of the complex numbers, we can show statement (7-2) is not satisfied by demonstrating it does not hold when we restrict our attention to the real numbers.  In that context, becomes the open interval (-1,1), and the inequality, , becomes , which for real variables is equivalent to .  If Statement (7-2) were to be satisfied, then given , would be within an -bandwidth of f(x) for all x in the interval (-1,1) provided n were large enough.  Figure 7.1 illustrates that there is an  such that, no matter how large n is, we can find such that lies outside this bandwidth.  In other words, Figure 7.1 illustrates the negation of Statement (7-2), which in technical terms we state as:  

(7-3)            There exists , such that for all positive integers N,
            there is some and some
            such that .  

In the exercises, we ask you to use Statement (7-3) to show that the partial sums of the geometric series do not converge uniformly to    for  .

Figure 7.1  The geometric series does not converge uniformly on (-1,1).  

Animation  The geometric series does not converge uniformly on (-1,1).   

Exploration.

    A useful procedure known as the Weierstrass M-test can help determine whether an infinite series is uniformly convergent.

Theorem 7.1.  (Weierstrass M-Test)  Suppose the infinite series    has the property that for each  k,  we have    for all  .  If    converges,  then    converges uniformly on   T.

Proof.

    Theorem 7.2 gives an interesting application of the Weierstrass M-test.

Theorem 7.2.  Suppose the power series    has radius of convergence  .  Then for each   r,  (where  ),  the series converges uniformly on the closed disk  .  

Proof.

    An immediate consequence of Theorem 7.2 is Corollary 7.1.

Corollary 7.1.  For each   r,  (where  ),  the geometric series    converges uniformly on the closed disk  .  

Proof.

    Theorem 7.3 gives important properties of uniformly convergent sequences.

Theorem 7.3.  Suppose is a sequence of continuous functions defined on a set   T  containing the contour   C.   
If    converges uniformly to    on the set  T,  then

(i)             is continuous on  T,  and  
    
(ii)          .   

Proof.

Corollary 7.2.  If the series    converges uniformly to    on the set  T,  and  C  is a contour contained in  T,  then  

        .   

Proof.

Example 7.2.  Show that      for all   .  

Solution.  For  ,  we choose r and R so that  ,  thus ensuring that    and that  .  By Corollary 7.1, the geometric series    converges uniformly to    on  .  If  C  is any contour contained in  ,  Corollary 7.2 gives  

(7-4)            .  

Clearly, the function    is analytic in the simply connected domain  ,  and    is an antiderivative of f(z) for all  ,  where Log is the principal branch of the logarithm.  Likewise,    is analytic in the simply connected domain  ,  and    is an antiderivative of g(z) for all  .  Hence, if C is the straight-line segment joining , we can apply Theorem 6.9 to Equation  (7-4) to get  

            ,  

which becomes

            ,  
            
which can be written as

            .   

The point was arbitrary, so we are done.

Explore Solution 7.2.