Module

for

Power Series Functions

 

4.4  Power Series Functions

    Suppose that we have a series [Graphics:Images/ComplexPowerSeriesMod_gr_1.gif]  where [Graphics:Images/ComplexPowerSeriesMod_gr_2.gif].  If [Graphics:Images/ComplexPowerSeriesMod_gr_3.gif] and the collection of [Graphics:Images/ComplexPowerSeriesMod_gr_4.gif] are fixed complex numbers, we will get different series by selecting different values for z.  For example, if [Graphics:Images/ComplexPowerSeriesMod_gr_5.gif] and [Graphics:Images/ComplexPowerSeriesMod_gr_6.gif] for all n, we get the series  [Graphics:Images/ComplexPowerSeriesMod_gr_7.gif] if [Graphics:Images/ComplexPowerSeriesMod_gr_8.gif],  and  [Graphics:Images/ComplexPowerSeriesMod_gr_9.gif] if [Graphics:Images/ComplexPowerSeriesMod_gr_10.gif].  Note that when  [Graphics:Images/ComplexPowerSeriesMod_gr_11.gif] and [Graphics:Images/ComplexPowerSeriesMod_gr_12.gif] for all n, we get a geometric series.  The collection of points for which the series [Graphics:Images/ComplexPowerSeriesMod_gr_13.gif] converges is the domain of a function [Graphics:Images/ComplexPowerSeriesMod_gr_14.gif], which we call a power series function.  Technically, this series is undefined if  [Graphics:Images/ComplexPowerSeriesMod_gr_15.gif] and n=0, since [Graphics:Images/ComplexPowerSeriesMod_gr_16.gif] is undefined.  We get around this difficulty by stipulating that the series [Graphics:Images/ComplexPowerSeriesMod_gr_17.gif] is really compact notation for [Graphics:Images/ComplexPowerSeriesMod_gr_18.gif].  In this section we present some results that are useful in helping establish properties of functions defined by power series.

Definition (Power Series).  The function  [Graphics:Images/ComplexPowerSeriesMod_gr_19.gif]  is called a power series, with center  [Graphics:Images/ComplexPowerSeriesMod_gr_20.gif].  

Theorem 4.15.  Suppose  [Graphics:Images/ComplexPowerSeriesMod_gr_21.gif] .  Then the set of points z for which the series converges is one of the following:

(i)    The single point  [Graphics:Images/ComplexPowerSeriesMod_gr_22.gif].  

(ii)   The disk  [Graphics:Images/ComplexPowerSeriesMod_gr_23.gif],  along with part (either none, or some or all) of the circle  [Graphics:Images/ComplexPowerSeriesMod_gr_24.gif].  

(iii)  The entire complex plane.

Proof.

Proof of Theorem 4.15 is in the book.
Complex Analysis for Mathematics and Engineering

 

    Another way to phrase case (ii) of Theorem 4.15 is to say that the power series [Graphics:Images/ComplexPowerSeriesMod_gr_25.gif] converges if [Graphics:Images/ComplexPowerSeriesMod_gr_26.gif] and diverges if  [Graphics:Images/ComplexPowerSeriesMod_gr_27.gif].  We call the number [Graphics:Images/ComplexPowerSeriesMod_gr_28.gif] the radius of convergence of the power series (see Figure 4.3).  For case (i) of Theorem 4.15, we say that the radius of convergence is zero and that the radius of convergence is infinity for case (iii).

                                   

                    Figure 4.3  The radius of convergence of a power series.  
                                       What happens on the boundary circle may be unknown.

Theorem 4.16 (Radius of Convergence).  For the power series function [Graphics:Images/ComplexPowerSeriesMod_gr_30.gif],  we can find  [Graphics:Images/ComplexPowerSeriesMod_gr_31.gif] , its radius of convergence, by any of the following methods:

    (i)    Cauchy's Root Test:   [Graphics:Images/ComplexPowerSeriesMod_gr_32.gif]     (provided that the limit exists.)  

    (ii)   Cauchy-Hadamard Formula:   [Graphics:Images/ComplexPowerSeriesMod_gr_33.gif]  (The limit superior always exists! see Definition 4.10 in Section 4.3)  

    (iii)  d'Alembert's Ratio Test:   [Graphics:Images/ComplexPowerSeriesMod_gr_34.gif]     (provided that the limit exists.)

We set  [Graphics:Images/ComplexPowerSeriesMod_gr_35.gif]  if the limit equals  0,  and  [Graphics:Images/ComplexPowerSeriesMod_gr_36.gif]  if the  limit equals  [Graphics:Images/ComplexPowerSeriesMod_gr_37.gif].  

Proof.

Proof of Theorem 4.16 is in the book.
Complex Analysis for Mathematics and Engineering

 

    We now give an example illustrating each of these cases.

Example 4.21.  The infinite series  [Graphics:Images/ComplexPowerSeriesMod_gr_38.gif]  has radius of convergence [Graphics:Images/ComplexPowerSeriesMod_gr_39.gif] Cauchy's root test because  

            [Graphics:Images/ComplexPowerSeriesMod_gr_40.gif],  

hence   [Graphics:Images/ComplexPowerSeriesMod_gr_41.gif].  

Explore Solution 4.21.

 

Example 4.22.   The infinite series  [Graphics:Images/ComplexPowerSeriesMod_gr_63.gif]  has radius of convergence [Graphics:Images/ComplexPowerSeriesMod_gr_64.gif] by the Cauchy-Hadamard formula.  We see this by calculating   

            [Graphics:Images/ComplexPowerSeriesMod_gr_65.gif],  so  

            [Graphics:Images/ComplexPowerSeriesMod_gr_66.gif],

hence  [Graphics:Images/ComplexPowerSeriesMod_gr_67.gif].  

Explore Solution 4.22.

 

Example 4.23.  The infinite series  [Graphics:Images/ComplexPowerSeriesMod_gr_97.gif]  has radius of convergence [Graphics:Images/ComplexPowerSeriesMod_gr_100.gif] by the ratio test because

            [Graphics:Images/ComplexPowerSeriesMod_gr_99.gif].

Since the limit equals  0,  we set  [Graphics:Images/ComplexPowerSeriesMod_gr_100.gif].    

Aside.  Just for fun, we can graph some approximations of this familiar complex exponential function.   

               [Graphics:Images/ComplexPowerSeriesMod.1_gr_1.gif]     [Graphics:Images/ComplexPowerSeriesMod.1_gr_2.gif]  
               [Graphics:Images/ComplexPowerSeriesMod.1_gr_3.gif]     [Graphics:Images/ComplexPowerSeriesMod.1_gr_4.gif]  
               [Graphics:Images/ComplexPowerSeriesMod.1_gr_5.gif]     [Graphics:Images/ComplexPowerSeriesMod.1_gr_6.gif]  

              The unit square  [Graphics:Images/ComplexPowerSeriesMod.1_gr_7.gif]  and it's images under the mappings:  [Graphics:Images/ComplexPowerSeriesMod.1_gr_8.gif],
               [Graphics:Images/ComplexPowerSeriesMod.1_gr_9.gif],    [Graphics:Images/ComplexPowerSeriesMod.1_gr_10.gif],    [Graphics:Images/ComplexPowerSeriesMod.1_gr_11.gif],    and    [Graphics:Images/ComplexPowerSeriesMod.1_gr_12.gif].  
               Remark. The accuracy of the image points for the approximation    [Graphics:Images/ComplexPowerSeriesMod.1_gr_13.gif]    is  
               [Graphics:Images/ComplexPowerSeriesMod.1_gr_14.gif].

Explore Solution 4.23.

 

Extra Example 1.  Find the radius of convergence of the infinite series  [Graphics:Images/ComplexPowerSeriesMod_gr_129.gif].

Explore Extra Solution 1.

 

 

    The Main Result of this section.

 

Theorem 4.17.  Suppose the function  [Graphics:Images/ComplexPowerSeriesMod_gr_178.gif]  has radius of convergence  [Graphics:Images/ComplexPowerSeriesMod_gr_179.gif].  Then

(i)    [Graphics:Images/ComplexPowerSeriesMod_gr_180.gif]  is infinitely differentiable for all [Graphics:Images/ComplexPowerSeriesMod_gr_181.gif].  In fact

(ii)   for all  k,   [Graphics:Images/ComplexPowerSeriesMod_gr_182.gif];   and

(iii)  [Graphics:Images/ComplexPowerSeriesMod_gr_183.gif]  where  [Graphics:Images/ComplexPowerSeriesMod_gr_184.gif]  denotes the [Graphics:Images/ComplexPowerSeriesMod_gr_185.gif] derivative of f.  (When [Graphics:Images/ComplexPowerSeriesMod_gr_186.gif],  [Graphics:Images/ComplexPowerSeriesMod_gr_187.gif] denotes the function [Graphics:Images/ComplexPowerSeriesMod_gr_188.gif] itself so that [Graphics:Images/ComplexPowerSeriesMod_gr_189.gif] for all z.)

Proof.

Proof of Theorem 4.15 is in the book.
Complex Analysis for Mathematics and Engineering

 

Example 4.24.  Show that   [Graphics:Images/ComplexPowerSeriesMod_gr_190.gif]   for all  [Graphics:Images/ComplexPowerSeriesMod_gr_191.gif]

Solution.  We know from Theorem 4.12 (in Section 4.3) that  [Graphics:Images/ComplexPowerSeriesMod_gr_192.gif]  for all  [Graphics:Images/ComplexPowerSeriesMod_gr_193.gif].  If we set k=1 in Theorem 4.16, part (ii), then

              [Graphics:Images/ComplexPowerSeriesMod_gr_194.gif],

for all  [Graphics:Images/ComplexPowerSeriesMod_gr_195.gif].

Explore Solution 4.24.

 

Extra Example 2.  Show that   [Graphics:Images/ComplexPowerSeriesMod_gr_202.gif]   for all  [Graphics:Images/ComplexPowerSeriesMod_gr_203.gif]

Explore Extra Solution 2.

 

Example 4.25.  The Bessel function [Graphics:Images/ComplexPowerSeriesMod_gr_210.gif] of order zero is defined by  

        [Graphics:Images/ComplexPowerSeriesMod_gr_211.gif],  

and termwise differentiation shows that its derivative is  

         [Graphics:Images/ComplexPowerSeriesMod_gr_212.gif]    

We leave as an exercise to show that the radius of convergence of these series is infinity.  The Bessel function [Graphics:Images/ComplexPowerSeriesMod_gr_213.gif] of order 1 is known to satisfy the differential equation  [Graphics:Images/ComplexPowerSeriesMod_gr_214.gif].

Explore Solution 4.25.

 

   [Graphics:Images/ComplexPowerSeriesFormulas_gr_1.gif]

                           Table.  Some power series and their known analytic formula.
                           Remark.  [Graphics:Images/ComplexPowerSeriesFormulas_gr_2.gif]  is a product of even numbers for n even, and odd numbers for n odd.

 

Looking Ahead

     In Section 6.5 we will prove that if f(z) is analytic at [Graphics:Images/ComplexPowerSeriesMod_gr_20.gif] then f(z) is infinitely differentiable at [Graphics:Images/ComplexPowerSeriesMod_gr_20.gif] (see Corollary 6.2).

     In Section 7.2 we will prove that if f(z) is analytic at [Graphics:Images/ComplexPowerSeriesMod_gr_20.gif] then f(z) has a Taylor series representation that is valid in some disk  [Graphics:Images/ComplexPowerSeriesMod_gr_23.gif] (see Theorem 7.4).

      Simply stated, if f(z) is complex differentiable in a neighborhood of [Graphics:Images/ComplexPowerSeriesMod_gr_20.gif] then it has a Taylor series representation.

 

Exercises Section 4.4.  Power Series Functions

 

 

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