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\centerline{\bf Class \#6 (Monday January 26)}
\begin{description}
\item[SUMMARY] Functions of a Complex Variable
\item[CURRENT READING] Brown \& Churchill, pages 26-32
\item[NEXT READING] Brown \& Churchill, pages 33-38
\end{description}
 
{\large\bf Functions of a Complex Variable}\\
Given a set $S$ of complex numbers, a function $f$ is a rule which assigns to each $z \in S$ a complex number $w$. The set $S$ is knows as the {\bf domain of definition} of $f$. (NOTE: The ``domain of definition'' is not necessarily a {`bf domain} in the formal mathematical sense of the word.) The set of all $w \in {\cal Z}$ given by $w = f(S)$ is called the {\bf image of $S$ under $f$} or the {\bf range of $f$.}\\
The value $w=f(z)$ can be written as $u+iv$, in other words:
$$w= f(z) = f(x+iy) = u(x,y) + i v(x,y)$$
where $u(x,y)$ and $v(x,y)$ are real functions of two real variables. 
 
\underline{\bf Example}\\
Write $f(z) = z^2 - z + 2i$ in the form $w = u(x,y) + i v(x,y)$
\vfill
 
In addition, given a function $w(x,y)$ you can write it in terms of $z$, $\zc$ and constants.

\underline{\bf Example}\\
Write $w(x,y) = x^2 + i y^2$ in terms of $z$ and $\zc$
\vfill

{\large\bf Mapping}\\
As usual, operations using complex variables have geometric significance.\\
First, let's get more practice evaluating functions of a complex variable:\\
Using $f(z) = z^3$, compute 
\begin{description}
\item[(a)] $f(2) = $
\item[(b)] $f(\sqrt{2}+i \sqrt{2}) = $
\item[(c)] $f(2i) = $
\end{description}

\newpage

\underline{\bf Example}\\
If you consider $w=f(z)$ a mapping from the $z$-plane to the $w$-plane, draw
a sketch of what the ``quarter-disc'' of radius 2 in the first quadrant of the 
$z$-plane maps to in the $w$-plane under the mapping $f(z) = z^3$.

In the language of mathematics, we say that what you draw in the $w$-plane is 
{\bf the image} of the quarter-disk in the $z$-plane {\it under} the mapping $f(
z)$

\begin{enumerate}
\item Write down a definition of the ``quarter disk of radius 2'' using complex 
nequalities
\item Shade in this region on your $(x,y)$ axes ($z$-plane) below
\item Shade in the mapped region on your $(u,v)$ axes ($w$-plane) below
\end{enumerate}
\begin{center}
\begin{tabular}{ll}
{\psfig{figure=fig/01.26-fig1.ps,height=2.5in,angle=270}} &
{\psfig{figure=fig/01.26-fig1.ps,height=2.5in,angle=270}}
\end{tabular}
\end{center}
On the other set of axes, sketch what the image of the mapping of the unit ``quarter-disk'' under $f(z)$ = $2 z^4 - 2 - i$ looks like in the $w$-plane.
\begin{center}
\begin{tabular}{ll}
{\psfig{figure=fig/01.26-fig2.ps,height=2.5in,angle=270}} &
{\psfig{figure=fig/01.26-fig2.ps,height=2.5in,angle=270}}
\end{tabular}
\end{center}

\newpage
We generally can decompose mappings into 3 dominant characteristics or components. That is, mappings can be described as some combination of {\bf rotation}, {\bf translation} and {\bf reflection}.

{\bf Rotation}\\
Consider $f(z) = (1+i)z$. How does this function represent a rotation mapping?\\ (You may want to consider its effect on the set of points Im $z = 0$.)
\begin{center}
\begin{tabular}{ll}
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}} &
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}}
\end{tabular}
\end{center}
\vfill

{\bf Reflection}\\
Consider $f(z) = \zc$. How does this function represent a reflection mapping?\\ (You may want consider the effect of $f$ on the set of points Im $z = 2$.)
\begin{center}
\begin{tabular}{ll}
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}} &
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}}
\end{tabular}
\end{center}
\vfill

{\bf Translation}\\
Consider $f(z) = 2z - i$. How does this function represent a translation mapping?\\ (You may want consider the effect of $f$ on the set of points $|z| = 1$.)
\begin{center}
\begin{tabular}{ll}
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}} &
{\psfig{figure=fig/01.26-fig2.ps,height=2.0in,angle=270}}
\end{tabular}
\end{center}
\vfill

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