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\begin{document}

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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 6.2-3  \\ Spring 2020 \hfill Due March 4}}}

\large

\vspace{1cm}

\begin{exercise}
Let $U, V \subset \C$ be domains, let $f: U \to \C$ be analytic with $f(U) \subset V$, and let $u : V \to \R$ be harmonic.  Prove that $u \circ f : U \to \R$ is harmonic.
\end{exercise}

\begin{exercise}
Let $z, w \in \C$.  Show that, when viewed as vectors in $\R^2$, one has
\[
z \cdot w = \frac{ \overline{z}w + z \overline{w} }{2} \,\,\, \text{ and } \,\,\, z \times w = \frac{\overline{z} w - z \overline{w}}{2i}.
\]
Conclude that the complex product can be expressed in terms of the dot and cross products:
\[
zw = \overline{z} \cdot w + i (\overline{z} \times w)
\]
\end{exercise}

\vspace{-0.5cm}

\begin{exercise}
Let $f = u + i v$ be an analytic function.  Let $k = u(z_0)$ and $\ell = v(z_0)$.  Prove that if $f'(z_0) \ne 0$, then the contours $u(x,y) = k$ and $v(x,y) = \ell$ are orthogonal at $z_0$.
\end{exercise}

\begin{exercise}
Find an expression for the cross-ratio $[z_0, z_1, z_2, z_3]$ when $z_j = \infty$, for $j=0,1,2,3$.
\end{exercise}

\begin{exercise}
Textbook exercise II.7.1.
\end{exercise}

\begin{exercise}
Textbook exercise II.7.9.
\end{exercise}


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