16.[A] Verify that whatever value is given to the complex number c, z = 1
is a solution of Fc(z) = 0, where Fc(z) =
17.[A] For the function Fc(z) =
(a) Verify that F'c(z) =
(b) Show zn+1 =
(c) Writing z = x + y*i and c = a + b*i, show that zn+1 in (b) has the following real and for the imaginary parts:
xn+1 =
and
yn+1 =
where
num1(n) =
num2(n) =
denom1(n) =
and denom2(n) =
18.[N] Using the iteration rules presented in exercise 16 (c), take a = 0.31, b = 1.65, x0 = 0, and y0 = 0. This point c = a + b*i lies in the main cardioid of the pseudo-Mandelbrot set in Figure 8.12. Using a calculator, compute x1, y1, ..., x14, y14. Deduce for this A + B*i the Newton method converges to a 2-cycle, not to a fixed point. Answer
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