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18.104
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Spring 2013
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\begin{center}\textbf{Homework 12}\end{center}

\begin{center}Due April 11th by the beginning of class.\end{center}

\textbf{Problem:} For $\lambda \in \mathbb C$ with $\textrm{Im}\ \lambda >0$, put 
\[
f(x) = (x^2 - \lambda^2)^{-1}.\]
 Prove that 
\[
\hat f (\xi) = \frac{i\pi}\lambda e^{i\lambda|\xi|},
\]
 and use the Poisson summation formula \cite[(8.5.3)]{frjo} to show that
\[
\sum_{k=-\infty}^\infty f(k) = \frac \pi \lambda \cot(\pi \lambda).
\]

\textit{Hint:} If you don't know the residue theorem from complex analysis, it might be easier to calculate the Fourier transform of $e^{i\lambda|x|}$ and use the Fourier inversion formula. The sum that results from Poisson summation reduces to a geometric series.

\textit{Hint 2:} The Poisson summation formula \cite[(8.5.3)]{frjo} implies that if $f \in \mathscr S(\mathbb R)$, then $\sum_{k=-\infty}^{+\infty} f(k) = \sum_{k=-\infty}^{+\infty} \hat f(2\pi k)$. In this problem you want to use this equation for a more general $f$. In fact, the equation is true as long as $f \in \mathscr S'(\mathbb R)$ is such that both $f$ and  $\hat f$ are continuous and there are constants $C>0$ and $\delta>0$ such that $|f(t)| + |\hat f(t)| \le C(1 + |t|)^{-1-\delta}$ for all $t$ in $\mathbb R$. For this problem you may use this version of Poisson summation without proof.

\textbf{Solution:}


\begin{thebibliography}{00}
\bibitem[FrJo]{frjo} G. Friedlander and M. Joshi. The Theory of Distributions, second edition, Cambridge University Press, 1998.
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