Module

for

Laurent Series Representations

 

7.3  Laurent Series Representations

    Suppose f(z) is not analytic in  [Graphics:Images/LaurentSeriesMod_gr_1.gif],  but is analytic in  [Graphics:Images/LaurentSeriesMod_gr_2.gif].   For example, the function  [Graphics:Images/LaurentSeriesMod_gr_3.gif]  is not analytic when  [Graphics:Images/LaurentSeriesMod_gr_4.gif]  but is analytic for [Graphics:Images/LaurentSeriesMod_gr_5.gif].  Clearly, this function does not have a Maclaurin series representation.  If we use the Maclaurin series for  [Graphics:Images/LaurentSeriesMod_gr_6.gif],  however, and formally divide each term in that series by  [Graphics:Images/LaurentSeriesMod_gr_7.gif], we obtain the representation  

    [Graphics:Images/LaurentSeriesMod_gr_8.gif]  

that is valid for all z  such that  [Graphics:Images/LaurentSeriesMod_gr_9.gif].  

 

Extra Example 1.  Use Mathematica to find the series for [Graphics:Images/LaurentSeriesMod_gr_10.gif].  

Explore Solution for Extra Example 1.

 

    This example raises the question as to whether it might be possible to generalize the Taylor series method to functions analytic in an annulus  

            [Graphics:Images/LaurentSeriesMod_gr_31.gif].   

Perhaps we can represent these functions with a series that employs negative powers of z in some way as we did with [Graphics:Images/LaurentSeriesMod_gr_32.gif].  As you will see shortly, we can indeed.  We begin by defining a series that allows for negative powers of z.  

 

Definition 7.3 (Laurent Series).  Let  [Graphics:Images/LaurentSeriesMod_gr_33.gif]  be a complex number for  [Graphics:Images/LaurentSeriesMod_gr_34.gif].  The doubly infinite series  [Graphics:Images/LaurentSeriesMod_gr_35.gif],  called a Laurent series, is defined by

(7-21)            [Graphics:Images/LaurentSeriesMod_gr_36.gif],  

provided the series on the right-hand side of this equation converge.

 

Remark 7.2.  Recall that  [Graphics:Images/LaurentSeriesMod_gr_37.gif]  is a simplified expression for the sum  [Graphics:Images/LaurentSeriesMod_gr_38.gif].  At times it will be convenient to write  [Graphics:Images/LaurentSeriesMod_gr_39.gif]  as  

            [Graphics:Images/LaurentSeriesMod_gr_40.gif],

rather than using the expression given in Equation (7-21).

 

Definition 7.4.  Given  [Graphics:Images/LaurentSeriesMod_gr_41.gif],  we define the annulus centered at  [Graphics:Images/LaurentSeriesMod_gr_42.gif]  with radii r and R by  

            [Graphics:Images/LaurentSeriesMod_gr_43.gif].  

The closed annulus centered at  with radii r and R is denoted by  

            [Graphics:Images/LaurentSeriesMod_gr_44.gif].
        
Figure 7.3 illustrates these terms.

 

Figure 7.3  The closed annulus [Graphics:Images/LaurentSeriesMod_gr_45.gif]. The shaded portion is the open annulus [Graphics:Images/LaurentSeriesMod_gr_46.gif].

 

Theorem 7.7.  Suppose that the Laurent series  [Graphics:Images/LaurentSeriesMod_gr_47.gif]  converges on an annulus  [Graphics:Images/LaurentSeriesMod_gr_48.gif].  Then the series converges uniformly on any closed subannulus  [Graphics:Images/LaurentSeriesMod_gr_49.gif] where  [Graphics:Images/LaurentSeriesMod_gr_50.gif].

Proof.

Proof of Theorem 7.7 is in the book.
Complex Analysis for Mathematics and Engineering

 

    The main result of this section specifies how functions analytic in an annulus can be expanded in a Laurent series.  In it, we will use symbols of the form  [Graphics:Images/LaurentSeriesMod_gr_51.gif],  which - we remind you - designate the positively oriented circle with radius  [Graphics:Images/LaurentSeriesMod_gr_52.gif]  and center  [Graphics:Images/LaurentSeriesMod_gr_53.gif].  That is,  [Graphics:Images/LaurentSeriesMod_gr_54.gif],  oriented counterclockwise.

 

Theorem 7.8  (Laurent's Theorem).  Suppose  [Graphics:Images/LaurentSeriesMod_gr_55.gif],  and that f(z) is analytic in the annulus [Graphics:Images/LaurentSeriesMod_gr_56.gif].  If  [Graphics:Images/LaurentSeriesMod_gr_57.gif]  is any number such that [Graphics:Images/LaurentSeriesMod_gr_58.gif], then for all [Graphics:Images/LaurentSeriesMod_gr_59.gif] the function value [Graphics:Images/LaurentSeriesMod_gr_60.gif] has the Laurent series representation

(7-22)            [Graphics:Images/LaurentSeriesMod_gr_61.gif],  

where for [Graphics:Images/LaurentSeriesMod_gr_62.gif], the coefficients [Graphics:Images/LaurentSeriesMod_gr_63.gif] are given by

(7-23)            [Graphics:Images/LaurentSeriesMod_gr_64.gif]    and    [Graphics:Images/LaurentSeriesMod_gr_65.gif].  

Proof.

Proof of Theorem 7.8 is in the book.
Complex Analysis for Mathematics and Engineering

 

Remark.  What happens to the Laurent series if f(z) is analytic in the disk [Graphics:Images/LaurentSeriesMod_gr_66.gif]?  If we look at equation (7-23), we see that the coefficient [Graphics:Images/LaurentSeriesMod_gr_67.gif] for the positive power [Graphics:Images/LaurentSeriesMod_gr_68.gif] equals [Graphics:Images/LaurentSeriesMod_gr_69.gif] by using Cauchy's integral formula for derivatives.  Hence, the series in equation (7-22) involving the positive powers of [Graphics:Images/LaurentSeriesMod_gr_70.gif] is actually the
Taylor series for f(z).  The Cauchy-Goursat theorem shows us that the coefficients for the negative powers of [Graphics:Images/LaurentSeriesMod_gr_71.gif] equal zero.  In this case, therefore, there are no negative powers involved, and the Laurent series reduces to the Taylor series.

 

    Theorem 7.9 delineates two important aspects of the Laurent series.

 

Theorem 7.9 (Uniqueness and differentiation of Laurent Series).  Suppose that  [Graphics:Images/LaurentSeriesMod_gr_72.gif]  is analytic in the annulus  [Graphics:Images/LaurentSeriesMod_gr_73.gif],  and has the Laurent series representation  

            [Graphics:Images/LaurentSeriesMod_gr_74.gif]  for all  [Graphics:Images/LaurentSeriesMod_gr_75.gif].  

(i)             If  [Graphics:Images/LaurentSeriesMod_gr_76.gif]  for all  [Graphics:Images/LaurentSeriesMod_gr_77.gif],  then [Graphics:Images/LaurentSeriesMod_gr_78.gif] for all  n.  
            (In other words, the Laurent series for f(z)  in a given annulus is unique.)

(ii)            For all [Graphics:Images/LaurentSeriesMod_gr_79.gif], the derivatives for [Graphics:Images/LaurentSeriesMod_gr_80.gif]  may be obtained by termwise differentiation of its Laurent series.

Proof.

Proof of Theorem 7.9 is in the book.
Complex Analysis for Mathematics and Engineering

 

    The uniqueness of the Laurent series is an important property because the coefficients in the Laurent expansion of a function are seldom found by using  Equation (7-23).  The following examples illustrate some methods for finding Laurent series coefficients.

 

Example 7.7.  Find three different Laurent series representations for the function  [Graphics:Images/LaurentSeriesMod_gr_81.gif]  involving powers of z.

    

[Graphics:Images/LaurentSeriesMod_gr_82.gif]

Solution.  The function f(z) has singularities at  [Graphics:Images/LaurentSeriesMod_gr_83.gif]  and is analytic in the disk  [Graphics:Images/LaurentSeriesMod_gr_84.gif],  in the annulus  [Graphics:Images/LaurentSeriesMod_gr_85.gif],  and in the region  [Graphics:Images/LaurentSeriesMod_gr_86.gif].  We want to find a different Laurent series for f(z) in each of the three domains D, A, and R.  We start by writing f(z) in its partial fraction form:  

(7-30)            [Graphics:Images/LaurentSeriesMod_gr_87.gif].  

We use Theorem 4.12 and Corollary 4.1 to obtain the following representations for the terms on the right side of Equation (7-30):  

[Graphics:Images/LaurentSeriesMod_gr_88.gif]

[Graphics:Images/LaurentSeriesMod_gr_89.gif]

[Graphics:Images/LaurentSeriesMod_gr_90.gif]

[Graphics:Images/LaurentSeriesMod_gr_91.gif]

[Graphics:Images/LaurentSeriesMod_gr_92.gif]

[Graphics:Images/LaurentSeriesMod_gr_93.gif]

[Graphics:Images/LaurentSeriesMod_gr_94.gif]

[Graphics:Images/LaurentSeriesMod_gr_95.gif]

[Graphics:Images/LaurentSeriesMod_gr_96.gif]

[Graphics:Images/LaurentSeriesMod_gr_97.gif]

[Graphics:Images/LaurentSeriesMod_gr_98.gif]

[Graphics:Images/LaurentSeriesMod_gr_99.gif]

  

Representations (7-31) and (7-33) are both valid in the disk  [Graphics:Images/LaurentSeriesMod_gr_100.gif],  and thus we have  

            [Graphics:Images/LaurentSeriesMod_gr_101.gif]   valid for   [Graphics:Images/LaurentSeriesMod_gr_102.gif],  

which is a Laurent series that reduces to a Maclaurin series.

In the annulus  [Graphics:Images/LaurentSeriesMod_gr_103.gif],  representations (7-32) and (7-33) are valid; hence we get  

            [Graphics:Images/LaurentSeriesMod_gr_104.gif]   valid for   [Graphics:Images/LaurentSeriesMod_gr_105.gif].  

Finally, in the region  [Graphics:Images/LaurentSeriesMod_gr_106.gif]   we use Representations (7-32) and (7-34) to obtain   

            [Graphics:Images/LaurentSeriesMod_gr_107.gif]   valid for   [Graphics:Images/LaurentSeriesMod_gr_108.gif].  

Explore Solution 7.7.

 

Example 7.8.  Find the Laurent series representation for  [Graphics:Images/LaurentSeriesMod_gr_221.gif]  that involves powers of z.  

Solution.  We know that  [Graphics:Images/LaurentSeriesMod_gr_222.gif],  and hence the Maclaurin series for  [Graphics:Images/LaurentSeriesMod_gr_223.gif]  is

            [Graphics:Images/LaurentSeriesMod_gr_224.gif],   
then we can write

            [Graphics:Images/LaurentSeriesMod_gr_225.gif]

or in another way we can write

            [Graphics:Images/LaurentSeriesMod_gr_226.gif]

We formally divide each term by [Graphics:Images/LaurentSeriesMod_gr_227.gif] to obtain the Laurent series  

            [Graphics:Images/LaurentSeriesMod_gr_228.gif]

Explore Solution 7.8.

 

Example 7.9.  Find the Laurent series for  [Graphics:Images/LaurentSeriesMod_gr_253.gif]  centered at  [Graphics:Images/LaurentSeriesMod_gr_254.gif].  

Solution.  The Maclaurin series for  exp z  is  [Graphics:Images/LaurentSeriesMod_gr_255.gif], which is valid for all z.  We let  [Graphics:Images/LaurentSeriesMod_gr_256.gif]  take the role of z in this equation to get

            [Graphics:Images/LaurentSeriesMod_gr_257.gif],

which is valid for [Graphics:Images/LaurentSeriesMod_gr_258.gif].  

Explore Solution 7.9.

 

Exercises for Section 7.3.  Laurent Series Representations

 

Library Research Experience for Undergraduates

Taylor Series  

Laurent Series  

Poles and Singularity  

 

 

  

 

The Next Module is
Singularities, Zeros and Poles

 

 

Return to the Complex Analysis Modules  

 

 

Return to the Complex Analysis Project

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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