% John David Stone
% Department of Mathematics and Computer Science
% Grinnell College
% stone@cs.grinnell.edu

% created June 21, 2002
% last revised June 21, 2002

\documentclass[11pt]{article}
\usepackage{amsmath}

\begin{document}
To evaluate the Green's function we impose a unit force on the $r$th
particle and we have to solve the equations
\begin{displaymath}
T_{n + 1}(\eta_{n + 1} - \eta_n) - T_n (\eta_n - \eta_{n - 1}) =
\begin{cases}0,  & n \neq r, \\
             -1, & n = r;
\end{cases}
\end{displaymath}
with $T_{N + 1} = 0$ and $\eta_0 = 0$ as the boundary conditions.  We
easily see (if only by ready appreciation of the physics) that for $r < n
\le N$, $\eta_n = \eta_r$.  Whereas for
\begin{displaymath}
1 \le n \le r, \qquad T_n(\eta_n - \eta_{n - 1}) = \text{constant}.
\end{displaymath}
Using the condition when $n = r$ we see that the constant is unity.  We
then have in turn
\begin{align*}
\eta_1 - \eta_0 &= 1/T_1, \\
\eta_2 - \eta_1 &= 1/T_2, \:\text{etc.} \\
\intertext{up to}
\eta_r - \eta_{r - 1} &= 1/T_r.
\end{align*}
Since $\eta_0 = 0$, adding the first $n$ equations,
\begin{displaymath}
\eta_n = 1/T_1 + 1/T_2 + \cdots + 1/T_n, \qquad 1 \le n \le r.
\end{displaymath}
Therefore the Green's function may be written
\begin{align*}
G(n, r) &= 1/T_1 + 1/T_2 + \cdots + 1/T_n, \qquad 1 \le n \le r, \\
        &= 1/T_1 + 1/T_2 + \cdots + 1/T_r, \qquad r \le n \le N;
\end{align*}
with $T_n = (N + n)(N - n + 1)$.
\end{document}