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\noindent
Math 436


\begin{center}
{\bf Take-home problems for final}
\end{center}

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\noindent
Due at 4:15 p.m., December 11, 2000

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\noindent
You may consult your course notes, homework, and the textbook.
You may use Maple, Matlab or Mathematica as an experimental
tool  on any part of the problem.  If you do, 
 please indicate how on your exam.
You may not consult any other books or notes.  You may not discuss
the exam with anyone except me.

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\noindent
The in-class part of the exam will be on December 11.  It will be
closed book, but you may bring either a sheet of paper ( $8\frac12 ''\times
11''\/$) with notes  or two
a  $3''\times 5''\/$
cards with notes to the exam.

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\noindent
You must justify your answers.
If you cannot do one part, you may assume the result of that 
part for the remaining parts.

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\noindent

1. (35 points) a)  Let  $u\/$
be a nonnegative harmonic function on  $\mathbf R^2\/$ (a solution of
Laplace's equation, $u_{xx} + u_{yy} = 0$).  Prove that
 $u\/$
is constant.
 
\medskip
 
({\bf Hint}: Let  $v_n(x,y) =u(nx,ny)\/$.  Show that  $v_n\/$
is harmonic.  Fix  $(x_0,y_0)\/$.  What does Harnack's inequality
tell you about $v_n(\frac{ x_0 }n,\frac{y_0}n)\/$
as  $n\to \infty\/$?  Harnack's inequality is the inequality
in \#24\,(a) on pp. 182-183.)
 
\medskip
 
\quad b) Prove {\bf Liouville's Theorem}:  Let  $u\/$
be a bounded harmonic function on  $\mathbf R^2\/$.  Then  $u\/$
is constant.
 
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2. (30 points)  A function  $g\/$
on  $\mathbf R^2\/$
is called  {\it radial }
if there is a function  $\psi\/$
on  $[0,\infty)\/$
such that  $g(x,y)=\psi\left(\sqrt {x^2+y^2}\,\right)\/$.
 Let  $f\/$
be an integrable function on  $\mathbf R^2\/$.  Suppose that
 $f\/$ is radial.  Show that the Fourier transform  $F\/$
of  $f\/$
is also radial.

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({\bf Hint}:  If  $\mathbf \mu  = ( \mu _1,\mu _2)\/$
and
$\widetilde{\mathbf \mu}  = (\widetilde \mu _1, \widetilde \mu _2)\/$
with  $|\mathbf \mu|=|\widetilde {\mathbf \mu}|\/$,
then there an angle  $\theta \/$
such that  $T_{\theta}(\mu)=\widetilde {\mathbf \mu}\/$
where  $T_{\theta}\/$
denotes rotation through angle  $\theta\/$.)

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3. (10 points) a) Did you find the portfolio assignment worthwhile?
Why or why not?

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\quad b) What suggestions do you have for improving the portfolio
assignment in the future.

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({\bf Note:} Full credit will be given for a thoughtful answer on 
this question.)

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