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\begin{document}

\title{Complex Analysis}
\author{Final Exam}
\date{December 20, 2021}
\maketitle

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\begin{enumerate}

\item                  
\begin{enumerate}
\item Find the real and imaginary parts of $\displaystyle
\frac{z+2}{z-1}$. 

\item Find all solutions to the equation $z^4 = 1$.

\item Find all complex solutions to the equation $\cos z = \sin z$.

\emph{Hint:} Let $t=\e^{\i z}$.

\item Find the radius of convergence of $\displaystyle
\frac{z^3-1}{z^2+3z-4}$ about the point $z=0$.

\end{enumerate}
\hfill(5+5+5+5)

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\item Short problems (I):

\begin{enumerate}

\item Is it true that $\lvert a^b \rvert = \lvert a \rvert^{\lvert b \rvert}$?

\item Suppose $f$ is entire with $\lvert f \rvert \geq 1$ on all of
$\C$.  Show that $f$ is a constant.

\item The function $f(z) = z^{-2}$ goes to zero as $\lvert z \rvert
\to \infty$.  Does this contradict Liouville's theorem?


\end{enumerate}
\hfill(5+5+5)
%\newpage

\item Short problems (II):

\begin{enumerate}

\item Show that $f(z) = 4z^6+z^2-2$ has all of its zeros inside the
unit disc.

\item Is
\[
  f(z) = \frac1{1+z^2}
\]
holomorphic on the exterior of the unit disc, $\{z \colon \lvert z
\rvert > 1\}$?

\item Let $f \colon \D \to \D$, where $\D$ denotes the open unit disc,
be holomorphic with $f(0)=0$.  Show that the power series expansion of
\[
  g(z) = \frac1{1-2 \, f(z)}
\]
converges on $\{z \colon \lvert z \rvert < \frac12 \}$.

\end{enumerate}
\hfill(5+5+5)
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\item
\begin{enumerate}
\item
Integrate the function
\[
  f(z) = \Re z
\]
around the boundary of the square $Q=[0,1] + \i [0,1]$ with standard
orientation.
\item Is $f$ holomorphic?
\end{enumerate}
\hfill(10+5)

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\item Let
$\gamma = \{ \zeta \colon \lvert \zeta \rvert = 1, \Im \zeta \geq 0
\}$ denote the upper-half unit circle endowed with standard
orientation.  Show that, for $\lvert z \rvert < 1$,
\[
  g(z) = \frac1{2\pi\i} \int_\gamma \frac{1}{\zeta - z} \, \d \zeta 
\]
is given by
\[
  g(z) = \frac12 + \frac{1}{2\pi\i} \log \frac{1+z}{1-z} \,.
\]
\emph{Note:} Be careful and explicit about the required branch cut.
\hfill(10)

%\newpage

\item Use contour integration to compute
\[
  \int_{-\infty}^{\infty} \frac1{x^4+1} \, \d x \,.
\]
\hfill (10)

%\newpage

\item Let $\Omega \subset \C$ be open and simply connected.  Show
that for any two $u,w \in \Omega$ there exists a conformal map $f
\colon \Omega \to \Omega$ such that $f(u)=w$.
\hfill(5)

%\newpage

\item Suppose that $f$ is holomorphic and of moderate decay on the
strip $\lvert \Im z \rvert < a$, i.e.,
\[
  \lvert f(x+\i y) \rvert \leq \frac{A}{1+x^2}
\]
for some $A>0$ and all $x \in \R$ and $\lvert y \rvert < a$.

Show that the Fourier transform of $f$,
\[
  \hat f (\xi) = \int_{-\infty}^\infty \e^{-2\pi\i x \xi} \,
  f(x) \, \d x
\]
satisfies the estimate
\[
  \lvert \hat f(\xi) \rvert \leq B \, \e^{-2 \pi  b \xi}
\]
for some $B>0$, every $0 < b < a$, and $\xi>0$.
\hfill(10)

\end{enumerate}
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\end{document}

