t} \,{\cal P}\int_0^{\pi} dk\,
\frac{1}
{\zeta+\cos
\frac{K}{2} \cos k}\nonumber\\
&&-\frac{i}{8t}\int_0^{\pi} dk\, \delta(\zeta+\cos
\frac{K}{2} \cos k).
\label{rint}\end{aligned}$$ Two cases must be distinguished: for $|\zeta| > |\cos(K/2)|$, the imaginary part of ${\cal R}$ is zero, and for $|\zeta| <
|\cos(K/2)|$ its real part is zero. The result is $${\cal R}(\zeta,K)=\left\{
\begin{array}{ll}
\displaystyle{\frac{1}{8t} \frac{\mathop{\rm sgn} (\zeta)}
{\sqrt{\zeta^2-\cos^2\frac{K}{2}}}}&
{}~~(|\zeta| > |\cos(K/2)|)\\
\\
\displaystyle{\frac{-i}{8t} \frac{1}{\sqrt{\cos^2\frac{K}{2}
-\zeta^2}}}&
{}~~(|\zeta| < |\cos(K/2)|).
\end{array}
\right.
\label{rintsecond}$$ The important quantity in equations (\[rhof\]) and (\[fdef\]) is $\arg {\cal R}$. From (\[rintsecond\]) we see that if $\zeta < -|\cos(K/2)|$, ${\cal R}$ is real and negative, while for $\zeta > |\cos(K/2)|$ it is real and positive. For $-|\cos(K/2)| \leq \zeta \leq |\cos(K/2)|$, ${\cal R}$ is pure imaginary with a negative imaginary part. So we find that $$\Phi(\zeta,K)=\mathop{\rm arg}\nolimits {\cal R}(\zeta,K)=\left\{
\begin{array}{ll}
-\pi & ~~(\zeta < -|\cos(K/2)|) \nonumber\\
-\pi/2 & ~~(|\zeta| \leq |\cos(K/2)|)\\
0 & ~~(\zeta > |\cos(K/2)|).\nonumber
\end{array}
\right.
\label{phi}$$ These values of $\Phi$ are plotted in figure \[fifig\] in the $\zeta,
K/2$ plane. Integrating in the $K$-direction, we find for $F(\zeta)$ $$\begin{aligned}
F(\zeta)&=&\frac{1}{N} \sum_K \mathop{\rm arg}\nolimits
{\cal R}(\zeta, K) = \frac{1}{2\pi}
\int_0^{2\pi} dK \Phi(\zeta,K) \nonumber \\
&&~~~= \left\{
\begin{array}{ll}
-\pi & ~~(\zeta < -1) \\
- \arccos \zeta & ~~(-1 \leq 