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%TCIDATA{<META NAME="Title" CONTENT="Plots of Complex Functions">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:13:21 1997}
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\section{\label{Conformal plot}Plots of Complex Functions}

A complex function $F(z)$ ($z$ and $F(z)$ are both complex) is a challenge
to graph, because the natural graph would require four dimensions.

\subsection{\ Conformal Plots}

A \emph{conformal plot} 
\index{Conformal plot@Conformal plot}of a complex function $F(z)$ is the
image of a two-dimensional rectangular grid of horizontal and vertical line
segments. The default is an $11$ by $11$ grid, with each of the intervals $%
0\leq $ Re$\left( z\right) \leq 1$ and $0\leq $ Im$\left( z\right) \leq 1$
subdivided into $10$ equal subintervals. If $F(z)$ is analytic, then it
preserves angles at every point at which $F^{\prime }(z)\neq 0$; hence, the
image is a grid composed of two families of curves that intersect at right
angles.

To create a conformal plot of $F(z)=%
\dfrac{z-1}{z+1}$, put the insertion point in the expression, and choose 
\textsf{Conformal} from 
\index{Plot 2D@Plot 2D!conformal@conformal}the \label{Plot 2D, conformal}%
\textsf{Plot 2D} submenu. The number of grid lines and the view can be
changed in the \textsf{Plot Properties} tabbed dialogs.

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Conformal}
\end{quote}

\begin{quotation}
$%
\dfrac{z-1}{z+1}$
\end{quotation}

\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language "Scientific
Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth 0pt;display
"PICT";plot-snapshots TRUE;function \TEXUX{$\dfrac{z-1}{z+1}$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "0";xmax
"1";ymin "0";ymax "1";xviewmin "-1.024";xviewmax "0.2245";yviewmin
"-0.02";yviewmax "1.02";phi 45;theta 45;plottype 9;num-x-gridlines
11;num-y-gridlines 11;numpts-xgridline 15;numpts-ygridline 15;plotstyle
"wireframe";axesstyle "normal";xis \TEXUX{z};var1name \TEXUX{$z$};boxtype
15;numboxes 15;}} In the following example, Re$(z)$ and Im$(z)$ both range
from $-3$ to $3$, the \textsf{View Intervals} are set at $-2\leq $ Re$\left(
F(z)\right) \leq 4$ and $-3\leq $ Im$\left( F(z)\right) \leq 3$, the \textsf{%
Grid Size} has been increased to $40$ by $40$, and \textsf{Samples per
Horizontal Grid Line} and \textsf{Samples per Vertical Grid Line} have both
been increased to $60$.$\medskip $

\begin{quote}
$\blacktriangleright $ \textsf{Plot 2D + Conformal}
\end{quote}

\begin{quotation}
$\dfrac{z-1}{z+1}$\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{}{\special{language
"Scientific Word";type "MAPLEPLOT";width 3in;height 2.0003in;depth
0pt;display "PICT";function \TEXUX{$\dfrac{z-1}{z+1}$};linecolor
"black";linestyle 1;linethickness 1;pointstyle "point";xmin "-3";xmax
"3";ymin "-3";ymax "3";xviewmin "-2";xviewmax "4";yviewmin "-3";yviewmax
"3";viewset"XY";rangeset"XY";phi 45;theta 45;plottype 9;num-x-gridlines
40;num-y-gridlines 40;numpts-xgridline 60;numpts-ygridline 60;plotstyle
"wireframe";axesstyle "normal";xis \TEXUX{z};var1name \TEXUX{$z$};boxtype
60;numboxes 60;}}
\end{quotation}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Vectors}{}{}{DM9-1.tex#Vectors}

\item  \hyperref{Plots of Vector Fields and Gradients}{}{}{DM9-3.tex#Vector
field}

\item  \hyperref{The View for Two Dimensional Plots}{}{}{DM6-3.tex#View
intervals}

\item  \hyperref{Two Dimensional Plots of Functions and Expressions}{}{}{%
DM6-4.tex#Plot expression}

\item  \hyperref{Complex Numbers}{}{}{DM2-5.tex#Complex numbers}

\item  \hyperref{Exercises and Solutions}{}{}{DM9-7.tex#Exercises}
\end{itemize}

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\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
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36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Complex numbers%
}{}{}{DMIndex.tex#Complex numbers}, \hyperref{Plotting 2D}{}{}{%
DMIndex.tex#Plotting 2D}, \hyperref{Plotting 3D}{}{}{DMIndex.tex#Plotting 3D}%
, \hyperref{Vector calculus}{}{}{DMIndex.tex#Vector calculus}}

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