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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 1.1  \\ Spring 2020 \hfill Due January 29}}}

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\vspace{1cm}


\begin{exercise}
If $z \in \C^{\times}$, prove that the equation $w^2 = z$ has exactly two solutions in $\C$.\footnote{That is, every nonzero complex number has exactly two square roots.}  Express $\operatorname{Re} w$ and $\operatorname{Im} w$ in terms of $\operatorname{Re} z$ and $\operatorname{Im} z$.
\end{exercise}

\begin{exercise}
For $x, y \in \R$, the {\em complex conjugate} of $z = x+i y \in \C$ is defined to be $\overline{z} = x - i y$.  Prove that the map $z \mapsto \overline{z}$ is a field automorphism of $\C$ with fixed field $\R$.\footnote{If $\sigma : F \to F$ is a field automorphism, the elements $a \in F$ satisfying $\sigma(a) = a$ form a subfield of $F$ called the {\em fixed field} of $\sigma$.}  What is its inverse?
\end{exercise}

\begin{exercise}
If $z \in \C^{\times}$, invert the matrix form of $z$ to find expressions for $\operatorname{Re}(1/z)$ and $\operatorname{Im}(1/z)$ in terms of $\operatorname{Re} z$ and $\operatorname{Im} z$.
\end{exercise}

\begin{exercise}
Show that the eigenvalues of the matrix form of $z \in \C$ are precisely $z$ and $\overline{z}$.
\end{exercise}

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