\nopagenumbers
\input amssym.def
\def\d{\displaystyle}
\font\tenbfsl =cmbxsl10 
\def\nn{$\hbox{\tenbfsl N}^{\,\bf 2}$}
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\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{\fiverm the last problem set $\scriptstyle \ldots$
no more problem sets $\scriptstyle \ldots$
that's it: we're done! $\scriptstyle \ldots$
the last problem set $\scriptstyle \ldots$
no more problem sets $\scriptstyle \ldots$
that's it: we're done! $\scriptstyle \ldots$}
\medskip

\centerline{\bf Homework \#7 \hfil \raise-1pt\hbox{\largerbold
Math 503} \hfil \bf November 29, 2004}
\medskip
\font\sevenit=cmti10 at 7pt
\font\seventt=cmtt10 at 7pt
\centerline{\bf Due Monday, December 13, 2004\quad}
%\centerline{\raise-.05pt\hbox{\largerbold Math 503}}
\parindent=0pt
\bigskip
\def\nn{$\hbox{\tenbfsl N}^{\,\bf 2}$}


The following three problems are from Remmert's {\sl Theory of Complex
Functions}. 

\medskip

Problem 1: For $a>1$ show that $\int_0^{2\pi} {{d\phi}\over{a+\sin
\phi}}= {{2\pi}\over{\sqrt{a^2-1}}}$.

\bigskip

Problem 2: Prove the identity $\int_{-\infty}^\infty
{{dx}\over{(x^4+a^4)^2}}= {3\over 8} {\sqrt{2}\over{a^7}}\pi$ for
$a>0$.  

\bigskip

Problem 3: Prove that $\int_0^\infty {{\sqrt{x}}\over {x^2+a^2}}
dx={\pi\over{\sqrt{2a}}}$. 

\bigskip

The following problem is Exercise 239 in \nn. Similar problems are
found on written qualifying exams of many universities. 

\medskip

Problem 4: For each positive integer $n$, and for each real
$\lambda>1$, prove that the equation $$z^n=e^{z-\lambda}$$ has {\it
no}\/ solutions with $|z|=1$, and exactly $n$ simple solutions with
$|z|<1$. 

\bigskip

A continuous function is {\it proper}\/ if and only if the inverse
image of every compact set is compact. The following problem is 
Exercise 297 in \nn.

\medskip

Problem 5: Prove that there is {\it no}\/ proper holomorphic map from
the open unit disc into the complex plane. 

\bigskip

The following problem is from Conway's {\sl Functions of One Complex
Variable}. 

\medskip

Problem 6: Does there exist a holomorphic function $f : D(0,1)\to
D(0,1)$ with $f\!\left({1\over 2}\right)={3\over 4}$ and
$f'\!\left({1\over 2}\right)={2\over 3}$?

\bigskip

The following problem is from Remmert's {\sl Theory of Complex
Functions}. Here $H$ is the open upper halfplane, so $H=\{z\in {\Bbb
C} : {\rm Im}\, z>0\}$.

\medskip

Problem 7: Let $f:D(0,1)\to H$ be holomorphic with
$f(0)=i$. If $f(0)=i$, prove that

\smallskip

a) ${{1-|z|}\over{1+|z|}}\le |f(z)| \le {{1+|z|}\over{1-|z|}}$ for
$z\in D(0,1)$.

\medskip

b) $|f'(0)|\le 2$.  

\bigskip

Problem 8: Let $f$ be a holomorphic function which maps the unit disk
into the unit disk. Show that $|f(z)+f(-z)|\le 2|z|^2$ for all $z$ in
the unit disk, and if the equality holds for some $z$, then
$f(z)=e^{i\theta}z^2$ for some real $\theta$.

\rightline{\sevenrm From a qualifying exam at Johns Hopkins
University\raise .2em\hbox{$\dag$}}

\vfil
\centerline{\fiverm the last problem set $\scriptstyle \ldots$
no more problem sets $\scriptstyle \ldots$
that's it: we're done! $\scriptstyle \ldots$
the last problem set $\scriptstyle \ldots$
no more problem sets $\scriptstyle \ldots$
that's it: we're done! $\scriptstyle \ldots$}

\eject

\font\show = cmdunh10.tfm
\font\sc=cmcsc10.tfm

\vfootnote{$\dag$}{This semester I've worked with a study group of
grad students who are preparing for our written exams. We looked at
this problem. At the urging of the {\sc very kind}\/ students in the
study group, I advise you that the problem statement, copied directly
from the Johns Hopkins exam, is incorrect. Please do one of the
following two alternative problems.

\smallskip

{\bf \# 1} Find a counterexample to the problem as stated. Then add a
simple hypothesis to the problem which makes it correct, and solve the
resulting problem.

\medskip

{\bf \# 2} Solve this problem, quoted from Remmert's  {\sl Theory
of Complex Functions}. Here $\Bbb E$ is the unit disc.\par\smallskip
{\parindent=.3in\narrower\noindent
Let $f :{\Bbb E}\to {\Bbb E}$ be holomorphic, with $f(0)=0$. Let $n\in
\Bbb N$, $n\ge 1$, $\zeta :=e^{2\pi i/n}$. Show that 
$$|f(\zeta z)+f(\zeta^2 z)+\cdots+f(\zeta^nz)|\le n|z|^n\quad {\rm
for\ all\ }z\in {\Bbb E}. \leqno{\qquad (*)} $$ 
\ \ Moreover, if there is at least one $c\in {\Bbb E}\setminus\{0\}$ such that equality
prevails in (*) at $z=c$, then there exists an $a\in\partial{\Bbb E}$
such that $f(z)=az^n$ for all $z\in {\Bbb E}$. {\it Hint.}\/ Consider
the function $h(z):={1\over{nz^{n-1}}}\sum_{j=1}^nf(\zeta^jz)$. For
the proof of the implication  
$f(\zeta z)+f(\zeta^2 z)+\cdots+f(\zeta^nz)=naz^n\Rightarrow f(z)=az^n$,
verify that the function $k(z):=f(z)-az^n$ satisfies
$$k(\zeta z)+k(\zeta^2 z)+\cdots+k(\zeta^nz)=0 \ \ {\rm and}\ \
\left|az^n\right|+2\Re(az^n\overline{k(\zeta^jz)})+\left|k((\zeta^jz)\right|^2<1$$
for every $j\in\{0,1,\ldots,n-1\}$, and consequently
$|k(z)|^2<n(1-|z|^{2n})$.  
\par}
 
}

\vfil\eject\end 
