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 \centerline{\large Math.\ 401, Sec.\ 500\hfill    Spring, 2005}
 \bigskip
\centerline{\bfseries\large Homework 10, due April 8}
\bigskip
                                           
                                       
\begin{enumerate}


\item {\it [Schaum's, p.\ 46, Ex.\ 2.46 and 2.48)]\/} \
Use Parseval's identity along with Exercise 5 of Homework 8 and 
Exercise 6 of Homework 9 to show (in any convenient order)

\begin{enumerate}
\item \ $\displaystyle \sum_{n=1}^\infty \frac1{n^4} = 
\frac{\pi^4}{90}\,$.

\item \ $\displaystyle \sum_{n=1}^\infty \frac1{n^6} = 
\frac{\pi^6}{945}\,$.

\item \ $\displaystyle \sum_{n=1}^\infty \frac1{(2n-1)^4} = 
\frac{\pi^4}{96}\,$.

\item \ $\displaystyle \sum_{n=1}^\infty \frac1{(2n-1)^6} = 
\frac{\pi^6}{960}\,$.
\end{enumerate}

\item {\it [Schaum's, p.\ 46, Ex.\ 2.56 and 2.57)]\/} \ 
\begin{enumerate}
\item A square plate of side $L$ has one side maintained at 
temperature $f(x)$ and the others at zero.
Find the steady-state temperature at any point of the plate
(as a Fourier series of appropriate type).
\item Explain how to solve the problem if the four sides 
are maintained at temperatures $f_1(x)$, $g_1(y)$, $f_2(x)$, and 
$g_2(y)$.
(Write out the answer in full for the case $f_2 =0=g_2\,$.)
\end{enumerate}

\item {\it [Schaum's, p.\ 46, Ex.\ 2.58(a))]\/} \
 An infinitely long plate of width $L$ has its two parallel sides 
maintained at temperature $0$ and its other side at constant 
temperature $T$.
Find the steady-state temperature.

\item {\it [Schaum's, p.\ 46, Ex.\ 2.63)]\/} \ 
Solve the boundary value problem
$$\frac{\partial u}{\partial t} = \frac{\partial^2u}{\partial 
x^2} - \alpha^2u, \qquad
u(0,t)=u_1\,, \quad u(L,t)=u_2\,, \quad
u(x,0)=0,$$
where $0<x<L$, $0<t$, and $\alpha$, $L$, $u_1\,$, and $u_2\,$  
are constants.

\smallskip
\noindent{\bf Instructions for Exercises 5 and 6:}
J. B. Fourier was ridiculed by some of the mathematicians of his day
when he first announced his discovery that an arbitrary function
on the interval $0<x<\pi$, such as $f(x) = x^2$, can be expanded
in a series of sine functions.
Some of the criticisms were like the two statements which follow.
In each case explain in a short essay how the mathematicians were 
confused (and Fourier was right).


\item ``$x^2$ is an even function;
but any fool can see that a sum of sines will always be an 
odd function.''

\item
``$x^2$ is not zero at the right endpoint ($\pi$);
but any fool can see that a sum of the functions
$\,\sin nx$ will always vanish at $x=\pi$.
The same criticism applies if we consider the limits of functions
as $x\to \pi$
instead of the values of the functions when $x=\pi$.'' 

\end{enumerate}
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