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\centerline{Worksheet \#3 (Friday 01/16/98)}
 
\begin{description}
\item[SUMMARY] Polar and Exponential forms of Complex Numbers
\item[READING] Churchill \& Brown, pages 12-18
\end{description}
We have been considering the complex plane as an analogue to the 2-D cartesian
x-y plane. You may recall that there are other coordinate systems that can be imposed on the 2-D plane. One of those coordinate systems is known as {\bf polar coordinates}.

\underline{\bf Exercise}\\
Write $(1,\sqrt{3})$ in polar coordinates.\\[24pt]
 What is the angle between a line drawn from $(0,0)$ to $(1,\sqrt{3})$ and the $x$-axis?\\[24pt]

The complex number which is found at $(1,\sqrt{3})$ is \underline{\hspace{1in}}.

It can also be written as $z = r ( \cos  \theta + i \sin \theta) = |z| \cis \theta$ \\
This is known as the {\bf polar form} of the complex number.

This angle $\theta$ corresponding to the complex number $z$ is called the {\bf principal argument} of $z,$ denoted by $\Arg z$. By tradition, $- \pi < \Arg z \le \pi$ and $\Arg 0$ is undefined.

The set of all values $\theta$  of such that ${\displaystyle\tan \theta = \frac{\Im z}{\Re z}}$ is known as the {\bf argument} of $z$, or $\arg z$

\begin{equation}
\arg z = \Arg z + 2 n \pi \qquad \qquad (n = 0, \pm1, \pm 2, \ldots)
\end{equation}

\underline{\bf Exercise}\\
Therefore, $\arg(1+\sqrt{3} i)$ is \underline{\hspace{3in}}.\\

Write $z_1 = -1-i$ and $z_2 = 1 - i$ in {\bf polar form} and sketch them on the graph below.

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Do you see any interesting geometrical relationship between $z_1$ and $z_2$?\\
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Do you notice any interesting algebraic relationship between $z_1$ and $z_2$?\\
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Write down a general relationship between arg$(z_1 z_2)$ and arg($z_1$) and arg$
(z_2)$\\
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\newpage
\centerline{\underline{\bf\Large Exponential Forms}}

{\bf Euler's Formula}\\
You may have noticed that the expression $\cos \theta + i \sin \theta $ is so common that the shorthand $\cis \theta$ has been developed. Another, even more compact symbol can be derived using Euler's formula:

$$e^{\displaystyle i \theta} = \cos \theta + i \sin \theta$$

Note $\theta = \arg z$ and is a real number in these formulas.
So, now we can write complex numbers in {\bf exponential form}
$$ z = |z| e^{\displaystyle i \theta} = r e^{\displaystyle i \theta}$$

\underline{\bf Exercise}\\
1. Compute the value of $e^{\pi i}$
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2. Write $1+i$ in exponential form
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3. Write $(1-i)^5$ in rectangular form (i.e. the usual $x+iy$)
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{\bf De Moivre's Formula}
$$(\cos \theta + i \sin \theta)^n = \cos n\theta + i \sin n\theta$$

We will prove this is true by first writing down $z^n$ in polar form and exponential form and equating them.
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