\documentclass {article}

\begin {document}

Please tell me something about the following expression:
\[
\rho (C) =
\frac {1} {2 \pi}
\int _ 0 ^ {2 \pi}
\cos \left [C \left (
\sum _ {i _ 1 = 1} ^ {N _ 1} c _ {i _ 1} \cos \left (\theta k _
{i _ 1} \right )
+
\sum _ {i _ 2 = 1} ^ {N _ 2} s _ {i _ 2} \sin \left (\theta l _
{i _ 2} \right )
\right ) \right ] d \theta
\]
$C \ge 0$;
$N _ 1, N _ 2 \in \{1, 2, \ldots, N\}$;
$c _ {i _ 1}, s _ {i _ 2} \in \{\pm 1, \pm 2, \ldots, \pm M\}$;
$k, l \in \{1, 2, \ldots, N\}$;
$k _ 1 < k _ 2 < \ldots < k _ {N _ 1}$; and
$l _ 1 < l _ 2 < \ldots < l _ {N _ 2}$.  All quantities are real
valued.

\begin {itemize}

\item
An upperbound on $\rho (C)$.
[\textit {\small {Hint: I think that $\rho (C) \le \sqrt {2 /
(\pi C)}$.  Note that $|H ^ {(1)} _ n (z)| \approx \sqrt {2 /
(\pi z)}$, $z \gg 1$, where $H ^ {(1)} _ n (z)$ is the
$n$th-order Hankel function of the first kind.}}]

\item
Show that $\rho (C) \le J _ 0 (C)$, $C \le x$, where $J _ 0 (z)$
is the 0th-order Bessel function of the first kind.
[\textit {\small {What is $x$?  $x \ll 1$? or something more
precise?}}]

\item
Show that $\lim _ {C \rightarrow \infty} \rho (C) = 0$.

\end {itemize}

Send your answers to Steve, sct@ucsd.edu.  One lucky winner gets
acknowledged in a forthcoming IEEE journal paper (co-authorship
is also a possibility).

[More hints: answers may lie in the \textit {generalized} Bessel
function.]

\end {document}