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\begin{document}

\centerline{\Large Homework 4: Zeros, Laurent, Residue theorem}
\vskip 1cm



\subsection*{Zeros}
\begin{enumerate}
        \item Show that the only holomorphic function $f$ on the unit disk such
          that $f(\frac 1n)=0$ for $n=2,3,\cdots$ is the zero
          function.
          \item Show that there are holomorphic functions $f$ other than
            the zero function on the punctured disk $\{z\mid |z|<1, z\neq 0\}$
            such that $f(\frac 1n)=0$ for $n=2,3,\cdots$. Also show
            that any such function has an essential singularity at 0.
            \item Show that there are no holomorphic functions $f$ on the unit disk such
          that $f(\frac 1n)=e^{-n}$ for $n=2,3,\cdots$.
          \item Show that the only holomorphic function $f$ on the unit disk such
          that $|f(\frac 1n)|\leq e^{-n}$ for $n=2,3,\cdots$ is the zero
          function. Hint: If not, write $f(z)=z^m g(z)$ with $g(0)\neq 0$.
          \item Suppose $f$ is holomorphic in $|z|<2$ and for every
            $n=2,3,\cdots$
            $$\int_{|z|=1} \frac {f(z)}{nz-1} dz=0$$
            Show that $f$ is the zero function.
            \item What can you say about $f$ if the displayed equation
              is replaced with
              $$\int_{|z|=1} \frac {f(z)}{(nz-1)^2} dz=0$$
            \item Is there a holomorphic function $f$ on the unit disk
              such that $$f\bigg(\frac 1{2n}\bigg)=f\bigg(\frac 1{2n+1}\bigg)=\frac 1n$$
              for $n=2,3,\cdots$?
              \item Suppose $f$ is an entire function and for every
                $z_0\in\C$ the power series expansion
                $$f(z)=\sum_{n=0}^\infty a_n(z-z_0)^n$$
                around $z_0$ has at least one coefficient $a_n$ equal
                to zero. Show that $f$ is a polynomial. Hint: Baire
                category theorem. 
\end{enumerate}

\subsection*{Laurent series}

\begin{enumerate}[resume]
\item Let $a,b\in\C$ with $0<|a|<|b|$ and let
  $$f(z)=\frac 1{(z-a)(z-b)}$$
  Find the Laurent series expansions of $f$ in
  \begin{enumerate}[(a)]
  \item $|z|<|a|$,
  \item $|a|<|z|<|b|$,
  \item $|z|>|b|$.
  \end{enumerate}

 \item Let $\alpha,\beta$ be two disjoint simple closed curves in $\C$ such
   that the disk $A$ bounded by $\alpha$ is contained in the disk $B$
   bounded by $\beta$. Let $\Omega$ be the domain (annulus)
   $int(B\smallsetminus A)$ and let $f:\Omega\to\C$ be
   holomorphic. Show that $f$ can be written uniquely as
   $$f=g+h$$
   where $g$ is holomorphic in $int(B)$, $h$ is holomorphic in
   $\C\smallsetminus \overline{A}$ and $h(z)\to 0$ as
   $z\to\infty$. This generalizes the case when $\alpha,\beta$ are
   concentric round circles, when $g$ corresponds to the part of the
   Laurent series with nonnegative powers of $z$.

\end{enumerate}

\subsection*{The Residue Theorem}

\begin{enumerate}[resume]
   
 \item Find the value of
   $$\int_{|z|=1}\sin\left(\frac 1z\right)~ dz$$

 \item Let $N$ be the year you were born. Compute
   $$\frac i4 \int_{|z|=N} \tan(\pi z)~dz$$

\end{enumerate}


  

\end{document}
