23. Show where some of the "good" (those that don't wind up at -infinity) values of x are in Figure 7.16. Convince yourself that these might form a Cantor set of some sort. Do not try to compute the dimension: this is not a standard Cantor set.

24.[A] Show equation (6) follows from (5). (Hint: use the quadratic formula.)

25.[A] Show that the fixed points of the Mandelbrot map (7) occur at

z_{f} = (1 ± sqrt(1 - 4*c))/2 Answer

26.[A] For what value of c is there a single fixed point?
What is z_{f} there? Answer

27.[N] Let w = 2 - 3*i and z = 4 + i. Compute

(a) w + z Answer

(b) w - z Answer

(c) w*z Answer

(d) w/z, being sure to put the answer into standard form. Answer

28.[N] Compute the magnitude of the answers to each part of exercise 27. Answer

29.[A] Show that the relation c^{2} = (a^{2} - b^{2})
+ (2*a*b)*i, where c = a + b*i, follows from the definition for complex multiplication.

30.[A] Show, for any complex numbers w and z, the magnitudes satisfy |w*z| = |w|*|z|.

31. Show that the complex numbers z lying on the unit circle satisfy |z| = 1. Show those lying inside the unit circle satisfy |z| < 1, and those outside satisfy |z| > 1.

32.[N] Show that the (filled-in) Julia set for c = 0 is the unit disc by completing
the following parts. (Observe, for c = 0, the Mandelbrot map (8) reduces to
z_{n+1} = z_{n}^{2}.)

(a) First, note that iteration of the Mandelbrot map produces the sequence
z^{2}, z^{4}, z^{8}, z^{16}, ... .

(b) Using a calculator with an x^{y} key, compute 1.1^{n}
for n = 2, 4, 8, 16, 32, 64, 128. Deduce if r > 1, then r^{2k}
runs away to infinity as k increases.

(c) Now compute 0.9^{n} for n = 2, 4, 8, 16, 32, 64, 128.
Deduce in 0 < r < 1, then r^{2k} goes to 0 an k increases.

(d) Using the answer to exercise 30, deduce that if the complex number z satisfies
|z| > 1, then |z^{2k}| runs away to infinity as k increases;
and if |z| < 1, then |z^{2k}| goes to 0 as k increases.

(e) If |z| = 1, deduce |z^{2k}| = 1 for all k.

(f) Deduce the Julia set for c = 0 consists of all z with |z| ² 1.

(g) Using exercise 31, identify the Julia set.

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