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\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Re{\operatorname{Re}}
\def\Im{\operatorname{Im}}

\begin{document}

\title{Complex Analysis}
\author{Homework 1}
\date{Due in class Wednesday, September 22, 2021}
\maketitle

\begin{enumerate}

\item Two fields $K$ and $L$ are isomorphic if there exists a
bijective map $\phi \colon K \to L$ such that
$\phi(x+y) = \phi(x)+\phi(y)$ and $\phi(xy) = \phi(x) \, \phi(y)$ for
all $x,y \in K$.

Show that $\C$ is isomorphic to the set of real $2\times 2$ matrices
of the form
\[
\begin{pmatrix}
  a & -b \\ b & a
\end{pmatrix}
\]
endowed with the usual matrix addition and matrix multiplication.

\item Derive the Cauchy--Riemann equations by referring to the
previous problem.

\emph{Hint:} Interpret $f \colon \Omega \subset \C \to \C$ as a map
between subsets of $\R^2$ and compute its Jacobian.  The CR-equations
now follow by inspection.  (Explain carefully why!)


\item Let $\Omega \subset \C$ be open and $f \colon \Omega \to \C$ be
holomorphic.  Set $\Omega^* = \{ \overline z \colon z \in \Omega \}$
and define $g \colon \Omega^* \to \C$ by
\[
  g(z) = \overline{f(\overline{z})} \,.
\]
Show that $g$ is holomorphic with
\[
  g'(z) = \overline{f'(\overline{z})} \,.
\]

\item Let $\Omega \subset \C$ be open and $f \colon \Omega \to \C$ be
holomorphic with $\Re f$ constant.  Prove that $f$ is a constant
function.

\item Determine the radius of convergence of the power series
$\sum_{n=1}^\infty a_n \, z^n$ when
\begin{enumerate}
\item $a_n = n^2$
\item $a_n = n!$
\item $a_n =  (n!)^3 /(3n)!$

\emph{Hint:} Use Stirling's formula, which says that
\[
  n! \sim \sqrt{2\pi n} \, \Bigl( \frac{n}\e \Bigr)^n 
\]
as $n \to \infty$.
\end{enumerate}


\end{enumerate}



\end{document}
