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\title{General Mathematics and ACM II}
\author{Exercise 14}
\date{April 1, 2011}
\maketitle

\begin{enumerate}
\item \emph{Extension from last exercise sheet:} Kac ring paper,
Exercise 12. 

\emph{Hints:}
\begin{enumerate}
\item Use Stirling's formula in the form
\[
  k! \sim \sqrt{2 \pi k} \, k^k \, e^{-k} \,.
\]
\item It is helpful, though not necessary, to take the logarithm of
the equation for the half-width early on.

\item Note that $\ln (n+w) = \ln n + \ln (1+w/n)$.  Then use the
log-series
\[
  \ln(1+x) = x - \frac{x^2}2 + O(x^3)
\]
to approximate such terms.  (All other logarithms, except the $\ln 2$
which survives into the final answer, should cancel identically.)

\end{enumerate}

\item Kac ring paper, Exercise 17. 
\item Kac ring paper, Exercise 19. 

(Use Matlab, Mathematica, or similar.  It would be nice to plot your
solution from Exercise 17 as well.)

\end{enumerate}
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