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Homework 1 problem set for Elec301 at Rice University.

Noon, Thursday, September 5, 2002

Homework, tests, and solutions from previous offerings of this course are off limits, under the honor code.

Form a study group of 3-4 members. With your group, discuss
and synthesize the major
*themes* of this
week of lectures. Turn in a one page summary of yourdiscussion. You need turn in only one summary per group,
but include the names of all group members. Please do notwrite up just a "table of contents."

Construct a WWW page (with your
*picture* ) and email Mike Wakin
(wakin@rice.edu) your name (as you want it to appear on theclass web page) and the URL. If you need assistance setting
up your page or taking/scanning a picture (both are easy!),ask your classmates.

Follow this learning styles link (also found on the Elec 301 web page ) and learn about the basics of learning styles. Write a short summary of what you learned. Also,complete the "Index of learning styles" self-scoring test on the web and bring your results to class.

Make sure you know the material in Lathi , Chapter B, Sections 1-4, 6.1, 6.2, 7. Specifically, be sureto review topics such as:

- complex arithmetic (adding, multiplying, powers)
- finding (complex) roots of polynomials
- complex plane and plotting roots
- vectors (adding, inner products)

Reacquaint yourself with complex numbers by going to the course applets web page and clicking on the Complex Numbers applet (may take a few seconds to load).

(a) Change the default add function to exponential (exp). Click on the complex plane to get a blue arrow, which isyour complex number $z$ . Click again anywhere on the complex plane to get a yellow arrow,which is equal to $e^{z}$ . Now drag the tip of the blue arrow along the unit circle on with $\left|z\right|=1$ (smaller circle). For which values of $z$ on the unit circle does $e^{z}$ also lie on the unit circle? Why?

(b) Experiment with the functions absolute (abs), real part (re), and imaginary part (im) and report your findings.

Reduce the following to the
Cartesian form,
$a+ib$ .
Do
*not* use your calculator!

(a) $\left(\frac{-1-i}{\sqrt{2}}\right)^{20}$

(b) $\frac{1+2i}{3+4i}$

(c) $\frac{1+\sqrt{3}i}{\sqrt{3}-i}$

(d) $\sqrt{i}$

(e) $i^{i}$

Find the roots of each of the following polynomials (show
your work). Use MATLAB to check your answer with the
`roots`

command and to plot the roots in the
complex plane. Mark the root locations with an 'o'. Putall of the roots on the same plot and identify the
corresponding polynomial (
$a$ ,
$b$ ,

(a) $z^{2}-4z$

(b) $z^{2}-4z+4$

(c) $z^{2}-4z+8$

(d) $z^{2}+8$

(e) $z^{2}+4z+8$

(f) $2z^{2}+4z+8$

$e^{\frac{i\times 2\pi}{N}}$ is called an Nth Root of Unity .

(a) Why?

(b) Let $z=e^{\frac{i\times 2\pi}{7}}$ . Draw $\{z, z^{2}, , z^{7}\}$ in the complex plane.

(c) Let $z=e^{\frac{i\times 4\pi}{7}}$ . Draw $\{z, z^{2}, , z^{7}\}$ in the complex plane.

A pair of vectors $u\in \mathbb{C}^{2}$ and $v\in \mathbb{C}^{2}$ are called linearly independent if $$u+v=0\text{if and only if}==0$$ It is a fact that we can write any vector in $\mathbb{C}^{2}$ as a weighted sum (or linear combination ) of any two linearly independent vectors, where the weights $$ and $$ are complex-valued.

(a) Write $\left(\begin{array}{c}3+4i\\ 6+2i\end{array}\right)$ as a linear combination of $\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\left(\begin{array}{c}-5\\ 3\end{array}\right)$ . That is, find $$ and $$ such that $$\left(\begin{array}{c}3+4i\\ 6+2i\end{array}\right)=\left(\begin{array}{c}1\\ 2\end{array}\right)+\left(\begin{array}{c}-5\\ 3\end{array}\right)$$

(b) More generally, write $x=\begin{pmatrix}x_{1}\\ x_{2}\\ \end{pmatrix}$ as a linear combination of $\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\left(\begin{array}{c}-5\\ 3\end{array}\right)$ . We will denote the answer for a given $x$ as $(x)$ and $(x)$ .

(c) Write the answer to (a) in matrix form,

(d) Repeat (b) and (c) for a general set of linearly independent vectors $u$ and $v$ .

A Julia set
$J$ is obtained by
characterizing points in the complex plane. Specifically,let
$f(x)=x^{2}+$ with
$$ complex, and define
$${g}_{0}(x)=x$$
$${g}_{1}(x)=f({g}_{0}(x))=f(x)$$
$${g}_{2}(x)=f({g}_{1}(x))=f(f(x))$$
$$$$
$${g}_{n}(x)=f({g}_{n1}(x))$$ Then for each
$x$ in the complex
plane, we say
$x\in J$ if the sequence
$$\{\left|{g}_{0}(x)\right|, \left|{g}_{1}(x)\right|, \left|{g}_{2}(x)\right|, \}$$ does
*not* tend to infinity. Notice
that if
$x\in J$ , then each element of the sequence
$\{{g}_{0}(x), {g}_{1}(x), {g}_{2}(x), \}$ also belongs to
$J$ .

For most values of $$ , the boundary of a Julia set is a fractal curve - it contains"jagged" detail no matter how far you zoom in on it. The well-known Mandelbrot set contains all values of $$ for which the corresponding Julia set is connected.

(a) Let $=-1$ . Is $x=1$ in $J$ ?

(b) Let $=0$ . What conditions on $x$ ensure that $x$ belongs to $J$ ?

(c) Create an approximate picture of a Julia set in MATLAB.
The easiest way is to create a matrix of complexnumbers, decide for each number whether it belongs to
$J$ , and plot the results
using the
`imagesc`

command. To
determine whether a number belongs to
$J$ , it is helpful to define
a limit
$N$ on the number of
iterations of
$g$ . For a
given
$x$ , if the magnitude
$\left|{g}_{n}(x)\right|$ remains below some threshold
$M$ for all
$0\le n\le N$ , we say that
$x$ belongs to
$J$ . The code
below will help you get started:

```
N = 100; % Max # of iterations
M = 2; % Magnitude threshold
mu = -0.75; % Julia parameter
realVals = [-1.6:0.01:1.6];
imagVals = [-1.2:0.01:1.2];
xVals = ones(length(imagVals),1) * realVals + ...
j*imagVals'*ones(1,length(realVals));
Jmap = ones(size(xVals));
g = xVals; % Start with g0
% Insert code here to fill in elements of Jmap. Leave a '1'
% in locations where x belongs to J, insert '0' in the
% locations otherwise. It is not necessary to store all 100
% iterations of g!
imagesc(realVals, imagVals, Jmap);
colormap gray;
xlabel('Re(x)');
ylabel('Imag(x)');
```

This creates the following picture for $=-0.75$ , $N=100$ , and $M=2$ .

Using the
*same* values for
$N$ ,
$M$ , and
$x$ , create a picture of the
Julia set for
$=-0.391-0.587i$ . Print out this picture and hand it in with yourMATLAB code.

Try assigning different color values to Jmap. For example,
let Jmap indicate the first iteration when the magnitudeexceeds
$M$ . Tip: try

`imagesc(log(Jmap))`

and
`colormap jet`

for a neat picture.-
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