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\noindent{\Large{\textsc{Complex Variables\hfill Assignment 5.2  \\ Spring 2020 \hfill Due February 26}}}

\large

\vspace{1cm}


\begin{exercise}
Let $a \in \C^{\times}$.  Prove that the principal branch of $f(z) = z^a$ is analytic on $\C \setminus (-\infty, 0]$ with $f'(z) = az^{a-1}$.
\end{exercise}

\begin{exercise}
Prove that, when extended to complex-valued functions, the differential operators $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are $\C$-linear and obey the product rule.  
\end{exercise}

\begin{exercise}
Use the preceding exercise to show that the operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline{z}}$ are $\C$-linear and obey the product rule.
\end{exercise}

\begin{exercise}
Let $\Omega \subset \C$ be a domain and let $D \! : \! C^1(\Omega) \to C^0(\Omega)$ be a $\C$-linear operator that obeys the product rule.  Prove that for any $f \in C^1(\Omega)$ and any $m \in \N$ one has $D(f^m) = mf^{m-1} D(f)$.  [{\em Suggestion:}  Induct on $m$.]
\end{exercise}

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