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\newcommand{\ds}{\displaystyle}
\newcommand{\pl}{\partial}
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\begin{document}


{\bf Question}

Consider the transformation $w=z^2.$  Sketch the curves in the
z-plane which map onto lines in the w-plane parallel to the real
and imaginary axis.

Prove that every straight line in the w-plane is the image of a
hyperbola (or pair of straight lines) in the z-plane.

\vspace{.25in}

{\bf Answer}

$z = x+iy \hspace{.3in} w = \alpha + i\beta$

$ \begin{array}{rcl} w = z^2 {\rm \ \ so\ \ } \alpha & = & x^2-y^2
\\ \beta & = & 2xy \end{array}$

$\beta$ = constant $\Leftrightarrow 2xy$ = constant

$\alpha$ = constant $\Leftrightarrow x^2-y^2$ = constant

\begin{center}
$\begin{array}{cc}
\epsfig{file=cn-17-1.eps, width=45mm} \ & \
\epsfig{file=cn-17-2.eps, width=45mm} \\
z = x+iy & w = \alpha + i \beta
\end{array}$
\end{center}

$\beta = m\alpha + c$ in $w$ plane $\Leftrightarrow$

$2xy = m(x^2 - y^2) + c$

$mx^2 - my^2 - 2xy + c = 0$

$\lq \lq b^2 - 4ac " = 4 + 4m^2 >0$ Therefore a hyperbola or $X$

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