7.5  Applications of Taylor and Laurent Series

    In this section we show how you can use Taylor and Laurent series to derive important properties of analytic functions.  We begin by showing that the zeros of an analytic function must be "isolated" unless the function is identically zero.  A point    of a set T is called isolated if there exists a disk    about    that does not contain any other points of T.  

Theorem 7.13.  Suppose f(z) is analytic in a domain D containing the point   and that  .   If f(z) is not identically zero, then there exists a punctured disk    in which f(z) has no zeros.

Proof.

Proof of Theorem 7.13 is in the book.
Complex Analysis for Mathematics and Engineering

    The proofs of the following corollaries are given as exercises.

Corollary 7.9.  Suppose that f(z) is analytic in the domain D,  and that  .  If there exists a sequence of points    such that  ,  and  ,  then    for all  .  

Proof.

Corollary 7.10.  Suppose that f(z) and g(z) are analytic in the domain D,  where  .  If there exists a sequence   such that  ,  and    for all  n,  then    for all  .  

Proof.

    Theorem 7.13 allows us to give a simple proof for one version of L'Hôpital's rule.

Corollary 7.11 (L'Hospital's Rule).  Suppose f(z) and g(z) are analytic at .  If    but ,  then  

            .   

Proof.

Proof of Corollary 7.11 is in the book.
Complex Analysis for Mathematics and Engineering

    We can use Theorem 7.14 to get Taylor series for quotients of analytic functions.  Its proof involves ideas from Section 7.2, and we leave it as an exercise.

Theorem 7.14  (Division of Power Series).  Suppose f(z) and g(z) are analytic at with power series representations  

               and      for all   .

If  ,  then the quotient    has the power series representation  

               for all   ,

where the coefficients satisfy the equations  .  

In other words, the series for the quotient    can be obtained by the familiar process of dividing the series for f(z) by the series for g(z) using the standard long division algorithm.

Proof.

Example 7.15.  Find the first few terms of the Maclaurin series for  ,  if  ,  and then compute .  

Solution.  Using long division, we see that  

              

              

              

              

              

              

Thus, we obtain   

            

Moreover, using Taylor's theorem, we see that if  ,  then     and we obtain  .

Explore Solution 7.15.

    We close this section with some results concerning the behavior of complex functions at points near the different types of isolated singularities.  Theorem 7.15 is due to the German mathematicia G. F. Bernhard Riemann (1826--1866).

Theorem 7.15  (Riemann's Theorem).  Suppose that f(z)  is analytic in  .  If f(z) is bounded in  ,  then either f(z) is analytic at    or f(z) has a removable singularity at .  

Proof.

Proof of Theorem 7.16 is in the book.
Complex Analysis for Mathematics and Engineering

    The proof of Corollary 7.12 is given as an exercise.

Corollary 7.12.  Suppose that f(z) is analytic in  ,  then f(z) can be defined to be analytic at    iff    exists and is finite.  

Proof.

Theorem 7.16.  Suppose that f(z) is analytic in  .  The function f(z) has a pole of order  k  at    iff  .  

Proof.

Proof of Theorem 7.16 is in the book.
Complex Analysis for Mathematics and Engineering

Theorem 7.17.  The function f(z) has an essential singularity at    iff    does not exist.

Proof.

Proof of Theorem 7.17 is in the book.
Complex Analysis for Mathematics and Engineering

Example 7.16.  Show that the function g(z) defined by  

                

is not continuous at  .

Solution.  In Exercise 20, Section 7.2, we asked you to show this relation by computing limits along the real and imaginary axes.  Note, however, that the Laurent series for g(z) in the annulus    is  

            ,

so that    is an essential singularity for g(z).  According to Theorem 7.17,    doesn't exist, so g(z) is not continuous at .

Explore Solution 7.16.