

\subsection{\bf (ii) Analysis of Scattering}
\subsection{\bf Experiments---I}

\begin{slide}
\begin{itemize}
\item Suppose a scattering experiment observes $\theta$ with
probability $p(\theta )$.

\item Then typically, one would expand $p(\theta) = {\displaystyle
\sum^\infty_{n=0}} a_n P_n (\cos\theta)$ as a series of Legendre
Polynomials $P_n (\cos\theta)$.

\item Then,
\begin{eqnarray*}
\left\langle P_{n_1} (\cos\theta)\right\rangle & = & \int\limits P_{n_1}
(\cos\theta) p(\theta) d\theta \\
& = & a_{n_1} \frac{2}{2n+1}
\end{eqnarray*}

\item Therefore, $a_n$ is proportional to $\left\langle P_{n_1}
(\cos\theta)\right\rangle$, which needs to be calculated as averages
over observed values, $\theta_k$.
\[
\left\langle P_n (\cos\theta)\right\rangle = \frac{1}{N} \sum^N_{k=1}
P_n \left( \cos\theta_k \right)
\]
\end{itemize}
\end{slide}

