\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
%\centerline{\bf \qquad Homework \#4 \hskip .55in \raise-1pt\hbox{\largerbold
%Math 503} \hskip .375in October 6, 2004}
\centerline{\bf Homework \#4 \hfil \raise-1pt\hbox{\largerbold
Math 503{$^\heartsuit$}} \hfil October 6, 2004}
\smallskip
\font\sevenit=cmti10 at 7pt
\font\seventt=cmtt10 at 7pt
\centerline{\bf Due Wednesday, October 20, 2004\ \ }
%\centerline{\raise-.05pt\hbox{\largerbold Math 503}}

\smallskip
\noindent Please finish reading \S 1.4 and \S 1.5 (pp.~32--43) in \nn. Then
we should begin chapter 2.

{\parindent=0pt

\vfootnote{$^{\heartsuit}$}{Thanks to Mr.~Nguyen for suggesting
these wonderful footnote characters.}

\medskip
Problem 1: Suppose that $f$ is holomorphic in $D(0,1)$ and that for
all sufficiently large integers, $n$, $\left|f\!\left(\!{1\over
n}\!\right)\right|\le {1\over{n!}}$. Prove that $f$ is the zero
function.

\medskip

Problem 2: a) An entire function is of {\it exponential type}\/ if
there are $A>0$ and $C>0$ so that for all $z \in\Bbb C$ with $|z|>C$,
$|f(z)|\le Ce^{A|z|}$. Prove that the collection of entire functions
of exponential type is closed under differentiation. In fact, if $f$
is of exponential type, so is $f'$, with the same $A$ but
with a possibly different $C$.

\smallskip

b) Show that an analogous statement for $C^\infty$ functions on $\Bbb
R$ is false. That is, define a $C^\infty$ real-valued function $g$ on
$\Bbb R$ to be of {\it exponential type}\/ if there are $A>0$ and
$C>0$ so that for all $x \in\Bbb R$ with $|x|>C$, $|f(x)|\le
Ce^{A|x|}$. Give an example of a $C^\infty$ function on $\Bbb R$ of
exponential type whose derivative is {\bf not} of exponential type for
{\it any}\/ choices of $C$ and $A$.$^{\textstyle \diamondsuit}$

\medskip

Problem 3: Suppose $U$ is open in $\Bbb R$. A function $f \colon U \to
{\Bbb R}$ is {\it real analytic}\/ or $C^\omega$ if, for every $a \in
U$, there is $\delta > 0$ with $(a-\delta, a+\delta) \subseteq U$ and
there is a sequence of real numbers $\{c_n\}$ so that $f(x) =
\sum\limits_{n=0}^{\infty} c_n (x-a)^n$ for all $x \in (a-\delta,
a+\delta)$: $f$ is $\underline{\rm real}$ analytic if $f$ is locally
the sum of a $\underline{\rm real}$ power series.

\smallskip

a) Suppose $F \colon \Omega \to {\Bbb C}$ is holomorphic and $F(\Omega
\cap {\Bbb R}) \subseteq {\Bbb R}$. If $U = \Omega \cap {\Bbb R}$ and
$f = F \! \bigm\vert_{U}$, prove that $f \colon U \to {\Bbb R}$ is
real analytic.

\smallskip

b) Suppose $f \colon U \to {\Bbb R}$ is real analytic with $U$ open
and connected in ${\Bbb R}$.  Prove that there is an open and
connected subset $\Omega$ of $\Bbb C$ and a complex analytic function
$F \colon \Omega \to {\Bbb C}$ so that $U = \Omega \cap {\Bbb R}$ and
$f = F \!  \bigm\vert_{U}$.\footnote{$^\clubsuit$}{$F$ is called a
{\it complexification}\/ of $f$.}

\smallskip

c) Find an $f$ which is $C^{\infty}$ on $\Bbb R$ but {\it not}\/
real analytic on $\Bbb R$.$^{\textstyle \diamondsuit}$ 


\smallskip

d) Find an $f$ which is real analytic on $\Bbb R$ whose Taylor series
at $0$ has finite radius of convergence.$^{\textstyle \diamondsuit}$

\smallskip

e) Liouville's Theorem is false for real analytic functions: find an
$f$ which is real analytic on $\Bbb R$, non-constant, and
bounded.$^{\textstyle \diamondsuit}$ \quad {\sevenrm Why does such an example
exist?}

\smallskip


f) An ``entire'' real analytic function may not have a
complexification with domain $\Bbb C$. If $\varepsilon>0$ define
$S_{\varepsilon}\subset\Bbb C$ by $S_{\varepsilon}=\{z \in{\Bbb C}:
|{\rm Im\,} z|<\varepsilon\}$. Construct a real analytic function $f
\colon {\Bbb R}\to {\Bbb R}$ so that there is {\it no}\/ holomorphic
$F_{\varepsilon}$ defined on $S_\varepsilon$ with
$F_{\varepsilon}\!  \bigm\vert_{\Bbb R}=f$ for {\it any}\/
$\varepsilon>0$.$^{\textstyle \diamondsuit}$



\vfootnote{$^{\textstyle \diamondsuit}$}{Easy examples are better than
complicated examples. I think there are easy examples for 2b), 3c), 3d), 3e), and 5b). Maybe there's an easy example for 3f) also. I would
like that.}

\medskip

Problem 4: a) Read a version of the Weierstrass Approximation
Theorem. The theorem statement should begin:
``If $f$ is continuous and real-valued on $[a,b]\subset \Bbb R$ and if
$\varepsilon > 0$, then there is $P(x)\in {\Bbb R}[x]$ so that
$\ldots$''

\smallskip

b) If $P(z)\in {\Bbb C}[z]$, prove that $\sup\{ \left|{1\over
z}-P(z)\right| : z\in \partial D(0,1)\}\ge 1$.

\vfil
\rightline{\bf OVER}
\vfil\eject

Problem 5: A holomorphic function $f$ has a {\it holomorphic log}\/ if
there is $g$, holomorphic in $f$'s domain, so that $e^g=f$.  A
holomorphic function $f$ has a {\it holomorphic square root}\/ if
there is $g$, holomorphic in $f$'s domain, so that $g^2=f$. Let
$D^*=\{z\in {\Bbb C} : 0<|z|<1 \}$.

\smallskip

a) If $f$ is holomorphic and has a holomorphic log, then
$f$ has a holomorphic square root. 

\smallskip

b) Give an example of a non-zero holomorphic function defined on $D^*$
which has a square root but does not have a log.$^{\textstyle
\diamondsuit\hbox{\ \sevenrm Previous\ page!}}$

\medskip

c) Prove that that $z$ has no holomorphic square root in
$D^*$.\footnote{$^\spadesuit$}{We did verify that $z$ has no log in
$D^*$ (if it did, then the log's derivative is ${1\over z}$ which has
no primitive in $D^*$). But because of b), a simple converse to a) is
not likely!}

\bigskip

Problem 6: a) Use what we've done so far in the course to find a
biholomorphic mapping of the first quadrant to the unit disc. What is
the image of the half line $\{y=x\}\cap\{x>0\}$? What
is the image of the quarter circle $\{|z|=1\}\cap\{0<{\rm arg\,}
z<{\pi\over 2}\}$?

\smallskip

b) Use what we've done so far in the course to find a biholomorphic
mapping of the strip $\{z\in {\Bbb C}: 0<{\rm Re}\,z<1\}$ to the unit
disc. What is the image of the line segment $ \{0<{\rm Re\,} z<1\}
\cap \{{\rm Im\,}z=0\}$? What is the image of the line
$\{{\rm Re\,}z={1\over 2}\}$?

}
\vfil\eject\end
  









