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\begin{document}

\centerline{\Large Review -- Problems from past quals}
\vskip 1cm

I didn't try to sort these in any way, some are very similar to each
other, and there are no hints (with one exception).

\begin{enumerate}[1.]

\item 
Let $f$ be a holomorphic function on a disk centered at 0 and radius
$>1$. Prove that if $|z|<1$ then
$$f(z)=\frac 1{2\pi i}\int_C \frac {f(\xi)}{\sin(\xi-z)} d\xi$$
where $C$ is the unit circle oriented counterclockwise.

\item Prove that if $f$ is an entire function satisfying
  $$|f(z)|\leq A+B\log |z|$$
  for some $A,B>0$ and all $z$ with $|z|\geq 1$, then $f$ is a
  constant function.

\item Determine if there exists an entire function $f$ such that
  $$f\bigg(\frac 1n\bigg)=f\bigg(-\frac 1n\bigg)=\frac 1{n^3}$$
  for all $n=1,2,\cdots$.

\item Let
  $$f(z)=\frac {e^{\frac 1{z-1}}}{e^z-1}$$
  Determine all isolated singularities of $f$ and their type. Also
  compute the residue at each pole.

\item Evaluate the integral
  $$\int_0^{2\pi}\frac{\cos^2 3\theta}{1-2a\cos\theta+a^2}d\theta$$
  where $a$ is a complex number with $|a|<1$.

  \item Let $f$ be a holomorphic function with an isolated singularity
    at $z=a$. Assume that $g=\frac 1f$ also has an isolated
    singularity at $z=a$, and that $a$ is not a removable singularity
    for either $f$ or $g$. Determine what type of singularity is $a$
    for $f$ and $g$. Do you know an example of such a function $f$?

    \item Let $f$ be holomorphic in $\C\smallsetminus
      \{0,2\}$. Assume:
      \begin{enumerate}[(i)]
      \item 0 and 2 are poles of order 1,
      \item $f$ is bounded on $|z|\geq 3$,
        \item the integral of $f$ over the circle $C(0,1)$ centered at
          0 and radius 1 is $2\pi i$, and
          \item the integral of $f$ over the circle $C(0,3)$ centered at
            0 and radius 3 is $0$.
      \end{enumerate}
      Determine $f$.

      \item Let $f$ be an entire function such that $f'=f$. Prove that
        $f(z)=Ce^z$ for some constant $C$. Hint: Power series

      \item Evaluate the integral
        $$\int_0^\infty \frac {\sin ax}{x(x^2+b^2)} dx$$
        where $a,b\in\R$ and $b\neq 0$.

      \item Characterize all entire functions $f$ such that
        $$\lim_{z\to\infty}\frac 1{f(z)}=0$$

        \item Let $\Omega\subseteq\C$ be open and $f$ holomorphic in
          $\Omega$. Let $b\in\Omega$ and assume $f'(b)\neq 0$. Show
          that
          $$\frac {2\pi i}{f'(b)}=\int_C \frac 1{f(z)-f(b)} dz$$
          for sufficiently small positively oriented circles $C$
          centered at $b$.

          \item Determine the poles and their orders of the function
            $$\frac 1{e^z-1}-\frac 1z$$

          \item Suppose $a>1$. Compute
            $$\int_0^{2\pi} \frac {d\theta}{a+\sin\theta}$$

          \item Evaluate
            $$\int_0^\infty \frac 1{x^3+1}dx$$
            
 \item Suppose $f$ is holomorphic in $0<r<|z|<R$ and suppose that for
   some $\rho$ with $r<\rho<R$
   $$\int_{|z|=\rho}f(z)z^n dz=0$$
   for all negative integers $n$. Show that $f$ extends to a
   holomorphic function on $|z|>r$.

   \item Let $f$ be an injective entire function and put
     $g(z)=f(1/z)$. Show that $g$ does not have an essential
     singularity at $z=0$. Further show that $f(z)=az+b$ for some
     constants $a,b$.

     \item Let $f$ be holomorphic on $D\smallsetminus\{0\}$ where $D$
       is the unit disk. Assume that $f'$ is bounded on
       $D\smallsetminus\{0\}$. Prove that $f$ extends to a holomorphic
       function on $D$.

       \item Let $f$ be an entire function such that $|f(z)|\neq 1$
         for all $z\in\C$. Show that $f$ is a constant function.

       \item Let $f$ be an entire function.
         \begin{enumerate}[(a)]
           \item If there is a polynomial $g$ such that $|f(z)|\leq
             |g(z)|$ for every $z$, show that $f$ is also a
             polynomial.
             \item Can we draw the same conclusion if $g$ is assumed
               to be a rational function instead of a polynomial?
         \end{enumerate}

         \item Show that any continuous function on $|z|\leq 1$ which
           is holomorphic on $|z|<1$ can be uniformly approximated by
           polynomials.

           \item Suppose $f$ is holomorphic on $|z|<1$ and satisfies
             $|f(1/n)|\leq 1/n^n$ for all $n=2,3,\cdots$. Determine
             $f$.

             \item If $f$ is entire and nowhere 0, show that there is
               an entire function $g$ such that $f=g^2$.

               \item If $f$ is entire and not a polynomial, show that
                 for every $\epsilon>0$ there is $z$ with
                 $|f(z)-z^2|<\epsilon$. 

\end{enumerate}


  

\end{document}
