
\subsection{\bf Some Mathematics Underlying}
\subsection{\bf Gaussian Integration}

\begin{slide}

\begin{itemize}
\item This can be ignored!

\item Let $P_n(x)$ be the polynomials orthogonal with respect
to weight function $g(x)$
$$
\int^b_a g(x)\,P_n(x)\,P_m(x)\,dx = h_n\, \delta_{nm} 
\eqno\hbox{(NI.10)}
$$
$h_n$ is a known irrelevant constant.  

\item Let $x^{(n)}_i$ be the $i\,$th zero of $P_n(x)$.  

\item Define $K_n(x,y)$ by the formula
$$
K_n(x,y) = \sum^n_{k=0}\, P_k (x)\, {P_k(y)\over{h_k}}
\eqno\hbox{(NI.11)}
$$

\item If $\Pi_n(x)$ is any polynomial of degree $n$, then one
can easily show:
$$
\int^b_a g(x)\, K_n(x,y)\, \Pi_n(x)\, dx = \Pi_n(y)
\eqno{\rm (NI.12)}
$$

\item It is not difficult to see that (NI.2) is satisfied by
\end{itemize}

$$
x_i = x_i^{(n+1)}\,, \hspace{1em}
w_i = {1\over{K_{n+1}\,\left( x_i^{(n+1)};\, x_i^{(n+1)}\right)}}
\eqno{\rm (NI.13)}
$$

\end{slide}

