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\centerline{\bf Class \#4 (Wednesday 01/21/98) }
\vspace{12pt}

{\bf SUMMARY} Polynomial Equations of a Complex Variable and Roots of Complex Numbers

{\bf CURRENT READING} Churchill \& Brown, pages 19-23

{\bf NEXT READING} Churchill \& Brown, pages 23-32
 
\underline{\bf Update}\\
The answers to the exercises from the class \#3 are:\\
{\bf 1:} $e^{\displaystyle i \pi} = \cos \pi + i \sin \pi = -1$\\
{\bf 2:} $1+ i = |1+i|e^{\displaystyle i \arg (1+i)} = \sqrt{2} e^{\frac{i \pi}{4} + 2 n i \pi}$\\
{\bf 3:} $(1-i)^5 =  \left( |1-i|e^{\displaystyle i \Arg (1-i)} \right)^5 = \left(\sqrt{2} e^{i \frac{- \pi}{4}} \right)^5 2^{5/2} e^{\frac{- i 5 \pi}{4}} = \sqrt{32} e^{\frac {3 \pi i}{4}} = \sqrt{32}\frac{(i-1)}{\sqrt{2}} = -4+4i$\\

We now know how to deal with real integer powers of complex numbers in a nice
way (by using DeMoivre's Formula and exponential form).\\ 

{\large\bf Fractional Exponents}\\
What about real fractional powers of complex numbers, i.e. roots?
That is, we want to solve an equation like 
\begin{equation}
z^{n} - z_0 = 0
\label{eq:root}
\end{equation}
where $z_0$ is a known complex number, and we are trying to find the corresponding value(s) of $z=z_0^{1/n}$ which solve this equation.

Suppose we write $z_0$, $z$ and $z^n$ in polar form:
\begin{eqnarray}
z_0 & =  & \\
z  &  =  & \\
z^n  &  =  &
\end{eqnarray}
where $|z_0| = r_0$, $|z| = r$, $\Arg z_0 = \theta_0$ and $\Arg z = \theta$

We can rewrite (\ref{eq:root}) as $z^n=z_0$ and obtain two real equations.\\

\underline{\bf Exercise}\\
Obtain expressions for $r$ and $\theta$ in terms of $r_0$ and $\theta_0$

\newpage
{\bf\large Roots of Unity}\\
We are interested in finding the $n^{th}$ roots of unty, i.e. $z$ such that
$$ z^n = 1, \qquad \qquad n = 1, 2, 3 \ldots$$

On the following axes, draw vector representations of the $n^{th}$ {\it roots of unity}
when $n =2$, $n=3$ or $n=4$. How many distinct solutions to $z^n=1$ are there?
In other words, we are trying to evaluate $\sqrt{1}$, $\sqrt[3]{1}$, $\sqrt[4]{1}$, $\ldots$

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What do you think the $5^{th}$ roots of unity will look like?\\
\underline{\bf Exercise}\\
Compute the solutions to the equation $z^5=i$ and write them in polar (and rectangular) form.  Sketch these solutions on the grid on the right.
\vfill

Using DeMoivre's Formula and the result on $n^{th}$ roots we can obtain a general formula for evaluating $z^{m/n}$
$$
\large
{\displaystyle z^{m/n} = c_k = |z|^{m/n} } \mbox{exp } {\displaystyle \left( \frac{i (\theta + 2 \pi k)}{n} \right)},  \qquad k = 0, 1, \ldots, n-1,\quad \mbox{where } \theta = \Arg z$$
\normalsize
These $n$ roots can be written as $$c,c \omega_n, c \omega_n^2, c \omega_n^3,\cdots, c \omega_n^{n-1}$$ where $c$ is {\it any} $n^{th}$ root of a non-zero complex number, and $\omega_n = \exp {\displaystyle \left( \frac{2 i\pi}{n} \right)}$

\newpage
\underline{Examples}\\
(1) Prove that $\sin 2\theta = 2 \sin \theta  \cos \theta $ and\\
 $\cos 2 \theta = \cos^2 \theta -  \sin^2 \theta$ (use DeMoivre's Formula)
\vfill
(2) Evaluate $\sqrt{5-12i}$
\vfill
(3) Solve $w^3 - i = -\sqrt{3}$
\vfill
(4) Solve $w^{4/3} + 2i = 0$
\vfill

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