\nopagenumbers
\input amssym.def
\def\d{\displaystyle}
\font\tenbfsl =cmbxsl10 
\def\nn{$\hbox{\tenbfsl N}^{\,\bf 2}$}
\magnification=\magstep1
\font\largerbold = cmbx10 scaled \magstep2
\font\largetbold = cmbx10 scaled \magstep1
%\centerline{\bf \qquad Homework \#5 \hskip .55in \raise-1pt\hbox{\largerbold
%Math 503} \hskip .375in October 6, 2004}
\centerline{\bf Homework \#6 \hfil \raise-1pt\hbox{\largerbold
Math 503} \hfil November 17, 2004}
\medskip
\font\sevenit=cmti10 at 7pt
\font\seventt=cmtt10 at 7pt
\centerline{\bf Due Monday, November 29, 2004\quad}
%\centerline{\raise-.05pt\hbox{\largerbold Math 503}}
\parindent=0pt
\bigskip\def\nn{$\hbox{\tenbfsl N}^{\,\bf 2}$}
Problem 1: Suppose $f$ is an entire function, and the Taylor series
centered at 0 
for $f$ converges uniformly to $f$ on $\Bbb C$. Prove that $f$ is a
polynomial.\footnote*{So uniform convergence on all of $\Bbb C$ is
{\bf wrong,} \raise-.15pt\hbox{\largetbold wrong,} \raise-.35pt\hbox{\largerbold wrong!}}

\bigskip

Problem 2.~Let $U$ be the region between the circles $|z-1|=1$ and
  $|z|=3$. Find a conformal mapping of an open neighborhood of
  $\overline{U}$ which carries $U$ to an annular region $A(0,r,R)=\{z
  : r<|z|<R\}$.

\bigskip

Problem 3.~Prove that for any fixed complex number $\zeta$, 
\smallskip
\centerline{$\d{1\over{2\pi}}\int_0^{2\pi} e^{2\zeta \cos
\theta}\,d\theta=\sum_{n=0}^\infty
\left({{\zeta^n}\over{n!}}\right)^{\! 2}.$}
\smallskip
\rightline{\sevenrm From a qualifying exam at the University of
California--Berkeley} 

\bigskip

Problem 4: Suppose $U$ is a simply connected open subset of $\Bbb C$,
and $f$ is a holomorphic function on $U$ which is not identically
$0$. Find a condition on the
zeros of $f$ which is equivalent to the existence of holomorphic
square root, $g$, of $f$ on $U$. Prove this
equivalence.\footnote{**}{We will analyze this situation when $f$ is
never 0 on $U$ in class. It is also discussed in section 2.5 of \nn.}

\bigskip 

Problem 5.~Let $D=\{z\in {\Bbb C} : |z|<1 \}$ be the open unit
disc. Find the image of $D$ under the map
$f(z)=e^{{i-iz}\over{z+1}}\,$.

\rightline{\sevenrm From a qualifying exam at Johns Hopkins University} 

\bigskip

Problem 6:~Suppose $f$ is holomorphic in the unit disk, and $f$ is not
identically $0$. Prove that there is a natural number $n$ such that
$f\!\left({1\over n}\right) \ne {1\over{n+2}}$.

\rightline{\sevenrm From a qualifying exam at the University of Iowa} 

\bigskip

Problem 7:~Suppose $S$ is the set $\{0\}\cup\{r=e^\theta :\theta\in
{\Bbb R}\}$ (the second set is an {\it equiangular spiral} described
in polar coordinates). $S$ is closed, and $U={\Bbb C}\setminus S$ is
open. Try to convince me that $U$ is simply connected. 

\hskip 1.9in{\hbox{\sevenrm 
\baselineskip=.75\baselineskip
\hsize=3.5in\vtop{Maybe this is part of what is ``fun'' in topology. 
Please do not spend too much time on this problem (it could take 10
pages, easily!). An attempted explanation would use the word compact
somewhere, and would sort of describe a homotopy in some
way. Sigh.\par}}}

\vfil\eject\end 


Problem 4:~Prove that the integral $\d\int_0^1 {{t^2}\over {t-z}}\,dt$
defines a holomorphic function $f(z)$ on ${\Bbb
C}\setminus[0,1]\,$. State Liouville's Theorem and use it to prove
that $f$ cannot be extended to $[0,1]$ in such a way to make $f$ an
entire function. %from where?

\bigskip






\vfil \eject \end


Gutzmer formula parseval, etc.

extend Liouville

u(x,y)&lt;A x^2 + B

cube roots and square roots imply logs!