140 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
(c) Use induction to show that if all the terms of the sequence {zk} are
defined, then the sequence {z•} is real, provided zo is real.
(d) Discuss whether {zk} converges to i if Im(zo) > 0 and to - i if
Im(zo) < 0.- Formulate and solve problems a nalogous to those in Exercise 1 for the function
l(z) = z^2 -l.
- P rove that Newton's method always works for polynomials of degree 1 (functions
of the form I (z) = az+b, where a f. 0). How many iterations are necessary before
Newton's method produces the solut ion z = -~ to I (z) = O? - Consider the function lo (z) = z^2 and an initial point zo. Let {zk} be the sequence
of iterates of zo generated by lo· That is, z 1 =lo (zo), z2 =lo (z1), and so on.
(a) Show that if lzol < l, the sequence {zk} converges to O.
(b) Show that if lzol > 1, t he sequence {zk} is unbounded.
(c) Show that if lzol = 1, the sequence {zk} either converges to 1 or oscil-
lates around the unit circle. Give a simple criterion that you can apply
to z 0 that will reveal which of these two paths { Zk} takes.- Show that the Julia set for l - 2 (z) is connected.
- Determine the precise structure of the set K-2·
- Prove that if z = c is in the Mandelbrot set, then its conjugate c is also in the
Mandelbrot set. Thus, the Mandelbrot set is symmetric about the z -axis. Hint:
Use mathematical induction.
8. Show that if c is any real number greater than i, then c is not in t he Mandelbrot
set. Note: Combining this condition with Example 4.11 shows that the cusp inthe cardioid section of the Mandelbrot set occurs precisely at c = i.
9. Find a value for c that is in the Mandelbrot set such that its negative, -c, is not
in the Mandelbrot set.- Show that the points c that solve the inequalities of T heorem 4.11 form a cardioid.
This cardioid is the main body of the Mandelbrot set shown in color plate 6. Hint:
It may be helpful to write the inequalities of T heorem 4.11 as
11. Use Theorem 4.11 and the paragraph immediately before it to show that the point- ~ J3i belongs to the Mandelbrot set.
- Suppose that {zo, z1} is a 2-cycle for f.
(a) Show that if zo is attracting for 92 (z), then so is the point z1. Hint:Differentiate 92 (z) = I(! (z)), using the chain rule, and show that
9Hzo) = 92(z1).
(b) Generalize part (a) ton-cycles.- Prove that k-oo lim Zk = zo in Theorem 4.10.