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\begin{document}
 \centerline{\large Math.\ 401, Sec.\
500\hfill    Spring, 2005}
 \bigskip
\centerline{\bfseries\large Homework 12,
due April 22}
\bigskip

\begin{enumerate}

\item {\it  [Schaum's, p.\ 94, Ex.\ 5.30]\/} \
Use Parseval's identity and your knowledge of the Fourier sine 
and cosine transforms of $e^{-x}$ ($x>0$) to evaluate

\begin{enumerate}
\item \ $\displaystyle \int_0^\infty \frac{dx}{(x^2+1)^2} $

\item \ $\displaystyle \int_0^\infty \frac{x^2\, dx}{(x^2+1)^2} $
\end{enumerate}

\item {\it [Schaum's, p.\ 94, Ex.\ 5.31]\/} \   
With the help of Exercise 2 from last week, show that
\begin{enumerate}
\item\ $\displaystyle \int_0^\infty \left({1-\cos x\over 
x}\right)^2
\,dx = 
\frac{\pi}2$

\item  \ $\displaystyle \int_0^\infty {\sin^2 x\over x^2}  \,dx = 
\frac{\pi}2$
\end{enumerate}

\item {\it [Schaum's, p.\ 94, Ex.\ 5.37]\/} \   
Find $y(x)$ given that
$$ \int_{-\infty}^\infty y(u)y(x-u)\,du = e^{-x^2}.$$

\item {\it [Schaum's, p.\ 94, Ex.\ 5.39]\/} \   
Prove that $f*(g*h) = (f*g)*h$.

 \item  Show that  differentiation of a Fourier transform with
respect to $\omega$ corresponds to multiplication of the original
function by $-ix$,
\begin{enumerate}
 \item starting from the equation
 $$g(x) = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty e^{i\omega 
x}\,\hat g(\omega)\, d\omega,$$
 applied to $\hat g(\omega ) = \hat f\,'(\omega)$;
 \item alternatively, starting from the equation
 $$\hat f(\omega)  = \frac1{\sqrt{2\pi}} \int_{-\infty}^\infty 
e^{-i\omega x}\, f(x)\, dx.$$
\end{enumerate}

 \item  Starting from the Fourier transform formulas in
exponential form (see previous problem, with $g=f$),
 derive the formulas for the Fourier sine transform.
 (Apply the Fourier transform formulas to the odd extension of $f$,
$f$ being an arbitrary nice function defined on $[0,\infty)$.)

\end{enumerate}
\end{document}