1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2 5 6 CHAP'l'ER 7 • TAYLOR AND LAURENT SERIES

QO

9. Consider the function ((z) = I:; n -', where n - • = exp(-zlnn).
   n = l

(a) Show that { (z) converges uniformly on the set A = {z: Re (z) 2'. 2}.

(b) Let D be a. closed disk contained in { z : Re ( z) > 1}. Show that < ( z)

converges uniformly on D.

7.2 Taylor Series Representations

In Section 4.4 we showed that functions defined by power series have derivatives

of all orders (Theorem 4.17). In Section 6.5 we demonstrated that analytic

functions also have derivatives of all orders (Corollary 6.2). It seems natural,

therefore, that there would be some connection between analytic functions and

power series. As you might guess, the connection exists via the Taylor and

Maclaurin series of analytic functions.

Definition 7 .2: Taylor series

If f (z) is analytic at z =a, then the series

J <2l(a) j <3l(o:)

J(a)+J'(a)(z-a)+

2

! (z- a)

2

+

3

, (z- a)

3

+···

00 J (k)()

= ~ a (z - a)k

L.J k!

k=O

is called the Taylor series for f centered at a. When the center i s a = 0,

the series is called the Maclaurin series for f.

To investigate when these series converge, we need Lemma 7.1.

t Lemma 7.1 If z, zo, and a are complex numbers with z =f z 0 and z =fa, then

1 I zo -a (zo - a)^2

• 
  
  
  
  • = --+ + ""----T
    z -ZO z - a (z - a)^2 (z -a)^3
    (zo - at 1 (zo - at+l
  
  
  
  

• · ' · + (z - a) n + l + z - zo (z - a) n+l '

where n is a positive integer.