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\begin{document}

\begin{center}
{\Large \em Math 560-61 (1998-99)}\\
\vspace{.3in}
{\Large Really Easy Takehome Final}\\
{due: Thursday, May 6, 1999.}
\end{center}

\large
\vspace{.2in}
\noindent
In the following questions the Fourier transform ${\cal F}$
of a function $u(x) \in L^2_c(\reals)$ and its inverse are defined via:
\begin{eqnarray}
{\cal F}(u)(\lambda) & \equiv & \hat{u}(\lambda) = 
 \int_{\reals} e^{i\lambda x} u(x) dx \ , \\
{\cal F}^{-1}(\hat{u})(x) & \equiv & u(x) = 
\frac{1}{{2\pi}}\int_{\reals} e^{-i\lambda x} \hat{u}
(\lambda) d\lambda \ .
\end{eqnarray}

\large
\begin{itemize}
\item[1.] Use the method of residues to find the inverse
Fourier transform of the function
\eq
\hat{u}(\lambda) =
\frac{1}{(\lambda^2+1)}
\endeq

\item[2.] 
For each
of the following, derive a formula for the Fourier transform of the 
indicated function (do not invert). In each question $x\in \reals$. 
(assume the problems are well posed).

\vspace{0.1in}
{\bf A} Find $\hat{u}(\lambda)$ in the following
\eq
\int_{\reals} f(x-y) u(y) dy +u(x) = g(x) \quad , \ 
\endeq

\vspace{0.1in}
{\bf B} Find $\hat{u}_2(\lambda)$ in the following
\begin{eqnarray}
u_1(x)+u_2(x) & = & a(x) \\
u_1(x-1)+2 u_2(x-1) & = & 0
\end{eqnarray}

\vspace{0.1in}
{\bf C} Find $\hat{u}(\lambda,t)$ in the following
\begin{eqnarray}
u_{tt}+u_{xxxx} & = & 0 \quad , \ u=u(x,t) \\
u(x,0) & = & f(x) \\
u_t(x,0) & = & 0
\end{eqnarray}

\end{itemize}

\end{document}