Chapter 9.6, Problem 45E

To determine

To graph: the first five members of the of each initial value using iteration method.

Answer to Problem 45E

  $z=-0.5-0.5i,z=-0.5+0.5i,z=0.5-0.5i,z=-0.5+0.5i\text{ and }z=0.5-0.5i.$

Explanation of Solution

Given:

  $\begin{array}{l}{z}_0=0.5-0.5i.\\ f\left(z\right)=z^2-\mathrm{0.5.}\end{array}$

Concept used:

Iteration:

An iterative sequence is one generated by the recurrence relation:

  $x_{n+1}=F\left(x_n\right).$

The starting value $x_0$ and each term is called an iterate.

In iterative method:

The order or the sequence of a function of preceding sequence or order.

Mathematically,

  $x_{n+1}=F\left(x_n\right).$

The graph of the iterative function can be drawn in Desmos graphing calculator.

Calculation:

First calculate the first five members of the orbit. Round the real and imaginary parts to the nearest hundredth.

  $\begin{array}{l}{z}_0=0.5-0.5i.\\ f\left(z\right)=z^2-\mathrm{0.5.}\end{array}$

First iteration:

  $\begin{array}{l}{z}_1=f\left(z_0\right).\\ \Rightarrow z_0{}^2-\mathrm{0.5.}\\ \Rightarrow {\left(0.5-0.5i\right)}^2-\mathrm{0.5.}\\ \Rightarrow 0.25-0.25-0.5i-0.5\\ z=-0.5-0.5i.\end{array}$

Second iteration:

  $\begin{array}{l}{z}_2=f\left(z_1\right).\\ \Rightarrow z_1{}^2-\mathrm{0.5.}\\ \Rightarrow {\left(-1\right)}^2{\left(0.5+0.5i\right)}^2-\mathrm{0.5.}\\ \Rightarrow 0.25-0.25+0.5i-0.5\\ z=-0.5+0.5i.\end{array}$

Third iteration:

  $\begin{array}{l}{z}_3=f\left(z_2\right).\\ \Rightarrow z_2{}^2-\mathrm{0.5.}\\ \Rightarrow {\left(-0.5+0.5i\right)}^2-\mathrm{0.5.}\\ \Rightarrow 0.25-0.25-0.5i-\mathrm{0.5.}\\ z=0.5-0.5i.\end{array}$

Fourth iteration:

  $\begin{array}{l}{z}_4=f\left(z_3\right).\\ \Rightarrow z_3{}^2-\mathrm{0.5.}\\ \Rightarrow {\left(-0.5-0.5i\right)}^2-\mathrm{0.5.}\\ \Rightarrow {\left(-1\right)}^2{\left(0.5+0.5i\right)}^2-\mathrm{0.5.}\\ z=-0.5+0.5i.\end{array}$

Fifth iteration:

  $\begin{array}{l}{z}_5=f\left(z_4\right).\\ \Rightarrow z_4{}^2-\mathrm{0.5.}\\ \Rightarrow {\left(-0.5+0.5i\right)}^2-\mathrm{0.5.}\\ \Rightarrow 0.25-0.25-0.5i-0.5\\ z=0.5-0.5i.\end{array}$

The sketch of iterates in complex plane is:

Hence, $z=-0.5-0.5i,z=-0.5+0.5i,z=0.5-0.5i,z=-0.5+0.5i\text{ and }z=0.5-0.5i.$