4.4  Power Series Functions

    Suppose that we have a series   where .  If and the collection of are fixed complex numbers, we will get different series by selecting different values for z.  For example, if and for all n, we get the series   if ,  and   if .  Note that when   and for all n, we get a geometric series.  The collection of points for which the series converges is the domain of a function , which we call a power series function.  Technically, this series is undefined if   and n=0, since is undefined.  We get around this difficulty by stipulating that the series is really compact notation for .  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    is called a power series, with center  .  

Theorem 4.15.  Suppose   .  Then the set of points z for which the series converges is one of the following:

(i)    The single point  .  

(ii)   The disk  ,  along with part (either none, or some or all) of the circle  .  

(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 converges if and diverges if  .  We call the number 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 ,  we can find   , its radius of convergence, by any of the following methods:

    (i)    Cauchy's Root Test:        (provided that the limit exists.)  

    (ii)   Cauchy-Hadamard Formula:     (The limit superior always exists! see Definition 4.10 in Section 4.3)  

    (iii)  d'Alembert's Ratio Test:        (provided that the limit exists.)

We set    if the limit equals  0,  and    if the  limit equals  .  

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    has radius of convergence Cauchy's root test because  

            ,  

hence   .  

Explore Solution 4.21.

Example 4.22.   The infinite series    has radius of convergence by the Cauchy-Hadamard formula.  We see this by calculating   

            ,  so  

            ,

hence  .  

Explore Solution 4.22.

Example 4.23.  The infinite series    has radius of convergence by the ratio test because

            .

Since the limit equals  0,  we set  .    

Explore Solution 4.23.

Extra Example 1.  Find the radius of convergence of the infinite series  .

Explore Extra Solution 1.

    The Main Result of this section.

Theorem 4.17.  Suppose the function    has radius of convergence  .  Then

(i)      is infinitely differentiable for all .  In fact

(ii)   for all  k,   ;   and

(iii)    where    denotes the derivative of f.  (When ,   denotes the function itself so that for all z.)

Proof.

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

Example 4.24.  Show that      for all  

Solution.  We know from Theorem 4.12 (in Section 4.3) that    for all  .  If we set k=1 in Theorem 4.16, part (ii), then

              ,

for all  .

Explore Solution 4.24.

Extra Example 2.  Show that      for all  

Explore Extra Solution 2.

Example 4.25.  The Bessel function of order zero is defined by  

        ,  

and termwise differentiation shows that its derivative is  

             

We leave as an exercise to show that the radius of convergence of these series is infinity.  The Bessel function of order 1 is known to satisfy the differential equation  .

Explore Solution 4.25.

                           Table.  Some power series and their known analytic formula.
                           Remark.    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  then f(z) is infinitely differentiable at  (see Corollary 6.2).

     In Section 7.2 we will prove that if f(z) is analytic at then f(z) has a Taylor series representation that is valid in some disk   (see Theorem 7.4).

      Simply stated, if f(z) is complex differentiable in a neighborhood of then it has a Taylor series representation.

Exercises Section 4.4.  Power Series Functions