\documentstyle[12pt,palatino,psfig]{article}
%\documentstyle[12pt]{article}
\topmargin=-1.25in
\oddsidemargin=0in
%\evensidemargin=0in
\textwidth=6.5in
\textheight=10.0in
\parindent=0in
%\parskip=10pt
\pagestyle{empty}

%\newfont{\cmss}{cmss12.tfm}
\tolerance=3000
%\setcounter{chapter}{1}

\begin{document}

\input 312top

 
\centerline{{\it  Class 15} {\bf (Wednesday February 18)}}
 
\begin{description}
\item[SUMMARY] Mappings by elementary functions
\vspace{-12pt}
\item[CURRENT READING] Brown \& Curchill, pages 75-81
\vspace{-12pt}
\item[NEXT READING] Brown \& Curchill pages 81-85
\end{description}

%\underline{\bf w = $\exp(z)$}
Consider the mapping consisting of the exponential function. Let's think about
where $w=e^z$ would map {\bf horizontal} and {\bf vertical} lines.

\underline{\bf Examples}\\
Let the vertical line be given by Re $z=\alpha$, in that case, the image of
this line is $w = e^{\alpha} \cdot e^{i y},$ where $y$ can be any real value.
What does this set of points look like? Sketch how the point $w$ changes as the pre-image point changes with $y$ [Hint: does $|w|$ depend on the size of $y$?]
%\vspace*{3in}
\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}
Now, figure out where $w=e^z$ maps horizontal lines and sketch them on the axes
above.\\

\underline{\sc Groupwork}\\
Putting this information together we can see that the set $0 < y < \pi \cap  x \le 0$ gets mapped to the interior of the semi-circle $(|w| < 1 \cap \Im w > 0$ under what mapping $w=f(z)$?
\vspace{1in}

How would we have to change our pre-image set so that $w=e^z$ would map the pre-image to the entire upper half-plane?
\newpage

\underline{\bf More mappings}\\
Now we know that the exponential function maps vertical and horizontal line segments to portions of circles (called {\it sectors}) and rays. The logarithm function is the inverse of the exponential function so we can use this information to determine its mapping properties.

As usual, we have to determine which {\it branch} of the logarithm we are talking about. For most purposes, we will just be interested in the principal branch, Log $z$. I usually think the best way to think about the logarithm mapping is
to try and think about the reverse of what the exponential function would do.

Here is an example of a useful mapping using $\Log z$ 
\vspace*{2in}


We say that a mapping is {\bf conformal} when it preserves the {\it angle} and {\it sense}  between two directed curves as it maps them from the $z$ plane to the $w$ plane. If the magnitude of the angle but {\it not the sense} is preserved, the mapping is called {\bf isogonal}
\vspace{1.5in}


\smallskip
Any analytic function which is nonconstant on a domain maps an open set to an open set. We will look at mappings by other analytic functions.
\end{document}
