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\centerline{Worksheet \#2 (Wednesday 01/14/98)}
 
\begin{description}
\item[SUMMARY] Graphical Representations of Complex Numbers and Inequalities
\item[READING] Churchill \& Brown, pages 6-11
\end{description}
Consider 2 complex numbers $z_1 = 3 +  0.5 i$ and $z_2 = 1 + 2i$\\
Draw an {\it Argand diagram} depicting these two complex numbers in the complex plane.
What physical quantity do $|z_1|$ and $|z_2|$ represent in the diagram?\\
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Then draw in vectors that represent the complex numbers $z_1+z_2$ and $z_1 - z_2$

Indicate what the value of $|z_1 + z_2|$ is. If I had two points at (3,0.5) and (1,2) what would the distance between these two points be?\\[24pt]

\underline{\bf Exercise}\\
Consider the equation $|z- 2 + i| = 2$.\\
What curve does this equation represent in the complex plane?\\[24pt]

Consider the equation $2 = \mbox{Re } (\bar{z} - i)$\\
What curve does this equation represent in the complex plane?\\[24pt]

Sketch the set of points which solve these equations on the grid provided.

\newpage
\centerline{\bf\large Complex Inequalities}
There are a number of interesting inequalities that one can prove using complex 
variables which have significance in other arenas. The most famous of these is the {\it Triangle Inequality}\\

{\bf Triangle Inequality}
\begin{equation}
| z_1 + z_2 | \le |z_1| + |z_2|
\label{ineq:tri}
\end{equation}

Can you express a geometric or vector interpretation of the Triangle Inequality?\\[24pt]

This expression (\ref{ineq:tri})  is also true in a more general form known 
as the {\it Cauchy-Schwarz Inequality}\\
{\bf Cauchy-Schwarz Inequality}
\begin{eqnarray}
 |z_1 + z_2 + z_3 + \ldots + z_n| &  \le  &|z_1| + |z_2| + |z_3| + \ldots + |z_n| \\
\left| \sum_{k=1}^{n} z_k \right| &  \le &\sum_{k=1}^{n} |z_k|
\end{eqnarray}
 
The Cauchy-Schwarz Inequality can be proved by {\it mathematical induction}.
\vspace{2in}

{\bf Some Other Inequalities}

For the inequalities below, try to  come up with a geometric interpretation and
prove that they are true.
$$\Re z \le | \Re z| \le |z| \mbox{ and } \Im z \le | \Im z | \le |z|$$

$$|z_1 + z_2| \ge ||z_1| - |z_2||$$

$$|z_1 - z_2| \le |z_1| + |z_2|$$

$$|z_1 - z_2| \ge ||z_1| - |z_2||$$

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