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\def\d{\displaystyle}
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%\centerline{\bf \qquad Homework \#5 \hskip .55in \raise-1pt\hbox{\largerbold
%Math 503} \hskip .375in October 6, 2004}
\centerline{\bf Homework \#5 \hfil \raise-1pt\hbox{\largerbold
Math 503} \hfil October 25, 2004}
\medskip
\font\sevenit=cmti10 at 7pt
\font\seventt=cmtt10 at 7pt
\centerline{\bf Due Wednesday, November 3, 2004\quad}
%\centerline{\raise-.05pt\hbox{\largerbold Math 503}}
\parindent=0pt
\bigskip
Problem 1: Suppose $f$ is holomorphic in the unit disc, and
$f\!\left({\sqrt{-1}\over{k}}\right)={{100}\over{k^4}}$ for integer
$k\ge 2$. What is $f$ exactly? 

\rightline{\sevenrm From a qualifying exam at the University of
California--Irvine} 

\bigskip

Problem 2: Let $f$ be a function defined in a domain $D$ in $\Bbb C$
such that $f(z)^3$ is holomorphic in $D$.

\smallskip

a) Is $f(z)$ holomorphic in $D$? (Prove or give a counterexample to
your answer.)\footnote{$^{\heartsuit}$}{\sevenrm This is how the
question was phrased. I think the sentence is weird.}

\medskip

b) If $f\in C^1(D)$, can you conclude that $f$ is holomorphic in $D$?
(Verify your answer.)


\rightline{\sevenrm From a qualifying exam at the University of
California--Irvine} 

\bigskip

Problem 3: 
Without using the Fundamental Theorem of Algebra, prove for any $P$ of
degree $n\ge 1$ that 
$$\lim_{R\to\infty}\int_{\partial D(0,R)} {{P'(z)}\over {P(z)}}\,dz =
2\pi i\, n.$$
Deduce the Fundamental Theorem of Algebra from this equality. 

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

\bigskip

Problem 4: Suppose that $U$ is an open subset of $\Bbb C$, and
$\{K_n\}_{n \in {\Bbb N}}$ is a sequence of compact sets so that
$\d\bigcup_{n\in {\Bbb N}} K_n =U$ and $K_n\subseteq {\rm
interior}(K_{n+1})$ for all $n\in {\Bbb N}$. Let $C(U)$ be the vector
space of continuous complex-valued functions defined on $U$. If $f$
and $g$
are in $C(U)$, define $d(f,g)$ by $$d(f,g)=\sum_{n=1}^\infty
{1\over {2^n}}{{\Vert f-g\Vert_{K_n}}\over {1+{\Vert
f-g\Vert_{K_n}}}}$$ where $\Vert h\Vert_S= \sup\{|h(z)| : z\in
S\}$. Prove that $(C(U),d)$ is a complete metric space, and that
$f_j\buildrel d \over \to f$ if and only if $\{f_j\}$ converges
uniformly to $f$ on every compact subset of $U$.

\bigskip

Problem 5: Find all uniformly continuous entire functions. 

\rightline{\sevenrm From a qualifying exam at Purdue University} 
\bigskip

Problem 6: Suppose that $f$ and $g$ are entire functions satisfying $|f(z)|\le
|g(z)|$ for all $z\in \Bbb C$. Prove that $f(z)=Cg(z)$ for some $C\in
\Bbb C$.

\rightline{\sevenrm This has appeared on written qualifying exams
given by at least three different schools.}

\bigskip

Problem 7: Suppose in this problem that $f$ has an isolated
singularity at 0.

\smallskip

a) Prove that if $e^f$ has a removable singularity at 0, then $f$
must have a removable singularity at 0. 

\medskip

b) Under what conditions does $e^f$ have a pole at 0?

\vfil \eject \end


Gutzmer formula parseval, etc.

extend Liouville

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



Suppose $f$ is an entire function. Prove that the Taylor series of $f$ at
0 converges to $f$ uniformly in $\Bbb C$ if and only if $f$ is a polynomial.
