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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 7.1  \\ Spring 2020 \hfill Due March 18}}}

\large

\vspace{1cm}

\begin{exercise}
Show that the FLT $f(z) = \frac{1+z}{1-z}$ maps the upper half of the unit disk \\ $\{ z \in \C \, | \, |z| < 1, \operatorname{Im} z > 0  \}$ conformally onto the first quadrant.
\end{exercise}

\begin{exercise}
Let $f(z) = \frac{z-i}{z+i}$.  What is the image under $f$ of
\begin{itemize}

\item[\bf a.]  the real line?

\item[\bf b.]  the circle with center 0 and radius 2?

\item[\bf c.]  the unit circle?

\item[\bf d.]  the imaginary axis?

\end{itemize}
\end{exercise}

\begin{exercise}
Find a conformal map that maps the quarter disk $\{ z \in \C \, | \, 0 < |z| < 1, 0 < \arg z < \pi/2  \}$ onto the open unit disk.  [{\em Suggestion:}  Compose several simpler maps.  Exercises 1 and 2 may be helpful.]
\end{exercise}

\begin{exercise}
Suppose $a, b, c, d \in \R$ and $ad - bc >0$.  Prove that $f(z) = \frac{az+b}{cz+d}$ maps the upper half-plane $H = \{ z \in \C \, | \, \operatorname{Im} z > 0\}$ onto itself.  [{\em Suggestion:}  Compute $\operatorname{Im} f(x+iy)$.]
\end{exercise}


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