Today we are going to be working with complex numbers. In Maple, we represent the complex number $i = \sqrt{-1}$ by the symbol

\[ e^x = 1 + \frac{x^1}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \]

\[ cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} + \cdots \]

\[ sin(x) = \frac{x^1}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

and so it is easy to see that

\[ e^{i x} = 1 + i \frac{x^1}{1!} - \frac{x^2}{2!} - i \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots = cos(x) + i sin(x) \]

A number is called complex if it is of the form $a + ib$ where $a$ and $b$ are real numbers. Any complex number represented as $a+ bi = r e^{ix}$ where $r = \sqrt{a^2 + b^2}$ and $x = arccos\left( \frac{a}{\sqrt{a^2+b^2}} \right)$.

Figure out what the following commands do.

> taylor( exp(x), x, 9 );

> subs(x=I*x,taylor( exp(x), x, 18));

> convert(subs(x=I*x,taylor( exp(x), x, 18)),polynom) - convert(taylor( cos(x), x, 18),polynom) - I*convert(taylor( sin(x), x, 18),polynom);

> I^2;

> exp(2*Pi*I);

> zeta3 := exp(2*Pi*I/3);

> zeta3 + zeta3^2 + zeta3^3;

> simplify(zeta3 + zeta3^2 + zeta3^3);

> [seq(expand(zeta3^r),r=0..12)];

> [seq(expand(1 + zeta3^r + zeta3^(2*r)),r=0..12)];

> Im(exp(2*Pi*I/6);

> sin(2*Pi/6);

> Re(exp(2*Pi*I/6));

> cos(2*Pi/6);

> with(plots);

> contourplot(x+y,x=-5..5,y=-5..5);

> plot3d(x+y,x=-5..5,y=-5..5);

> f:=z->Re(z)+Im(z);

> contourplot(f(x+I*y),x=-5..5,y=-5..5);

One of the examples above is a

Step 4: For the functions $g(z) = 1/z$ plot $f_1(g(z))$ and $f_2(g(z))$ and display the plots on the same graph. Explain what you see and compare your answer to the graphs on the help page for

For $z$ a complex number, let $g(z) = \frac{1+2z}{1+z}$. Find a function $h_1(z)$ such that $h_1(g(z)) = z$ and another (possibly the same) $h_2(z)$ such that $g(h_2(z)) = z$. Use Maple to simplify your expressions and show that your answers are correct. Explain your steps.

You should open up a new worksheet and start from scratch. You will have to save your work in a file and upload that file on to the course moodle. Your solution should be a sequence of commands where it is easy to change the input string and after you execute the sequence of commands you should have the correct output string. Please add documentation to your worksheet to explain how it works. Just a few sentences is sufficient, but imagine that you were opening up the worksheet for the first time and wanted to know what it did. You will be marked down if what you write is not clear and coherent.

You should finish your assignment before class Tuesday October 9 by 11:59pm. Assignments submitted after this date will be assessed a penalty of 10% per day.