1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

132 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES

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9. Show that, if E z., converges, then Jim z,.. = O. Hint: z,. = S.,. - $.,.._,.
   n=l n-oo


10. State whether the following series converge or d iverge. Just ify your answers.

(a) I: !!f.

n = l

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(b) E (~ + 2 ~) ·

n = l

00

11. 
    

    Let E (x,, + iy,,) = u + iv. If c = a + ib is a complex constant, show that
    n=l
    00
    E (a+ ib) (x,. + iy,..) =(a+ ib)(u +iv).
    n=l

    
    


12. 
    

    If n~O z,.. converges, show that L~o Znl S: n~o lznl·

    
    


13. 
    

    Complete the proof of T heorem 4 .1. In other words, suppose that n-oo Jim z,.. = (,
    where z,, = Xn + iy,, and ( = u +iv. Prove that n~oo Jim Yn = v.

    
    


14. 
    

    A side comment asked you to justify the first inequality in the proof of Theorem
    4.1. Give a justi6ca.tioo.
    1 5. Prove that a sequence can have only one limit. Hint: Suppose that there is a
    sequence {z,..} such that z., --+ (1 and z,.. --+ (2. Show t his implies (1 = (2 by
    proving that for all c: > 0, I< 1 - (2 I < c:.

    
    


15. 
    

    Prove Corollary 4.1.
    1 7. P rove that Jim z,. = 0 iff lim lz... I = 0.
    n -oo n -oo

    
    

4.2 Julia and Mandelbrot Sets

An impetus for studying complex analysis is the comparison of properties of

real numbers and functions with their complex counterparts. In this section we

take a look at Newton's method for finding solutions to the equation f (z) = 0.

Then, by examining the more general topic of iteration, we will plunge into a

breathtaking world of color and imagination. The mathematics surrounding this

topic has generated a great deal of popular attention in the past few years.

Recall from calculus that Newton's method proceeds by starting with a func-

tion f (x) and an initial "guess" of xo as a solution to f (x) = 0. We then generate

a new guess x 1 by the computation x 1 = xo -J ,«:~). Using x 1 in place of xo, this

process is repeated, giving x2 = x1 - J ,~·.». We thus obtain a sequence of points

{xk}, where Xk+i = xk - j~(;~). The points {xk}~ 0 are called the iterates

of xo. For functions defined on the real numbers, this method gives remarkably

good results, and the sequence {xk} often converges to a solution off (x) = 0

rather quickly. In the late 180 0s, the British mathematician Arthur Cayley in-

vestigated the question of whether Newton's method can be applied to complex

functions. He wrote a paper giving an analysis for how this method works for