Starting with expression (13.3.8) for the partition function of a one-dimensional \(n\)-vector model, with \(J_{i}=n J^{\prime}\), show that

\[
\operatorname{Lim}_{n, N \rightarrow \infty} \frac{1}{n N} \ln Q_{N}(n K)=\frac{1}{2}\left[\sqrt{ }\left(4 K^{2}+1\right)-1-\ln \left\{\frac{\sqrt{ }\left(4 K^{2}+1\right)+1}{2}\right\}\right]
\]

where \(K=\beta J^{\prime}\). Note that, apart from a constant term, this result is exactly the same as for the spherical model; the difference arises from the fact that the present result is normalized to give \(Q_{N}=1\) when \(K=0\).

Use the asymptotic formulae (for \(v \gg 1\) )

\[
\Gamma(v) \approx(2 \pi / v)^{1 / 2}(v / e)^{v}
\]

and

\[
I_{u}(u z) \approx(2 \pi u)^{-1 / 2}\left(z^{2}+1\right)^{-1 / 4} e^{u \eta},
\]

where

\[
\left.\eta=\sqrt{ }\left(z^{2}+1\right)-\ln \left[\left\{\sqrt{ }\left(z^{2}+1\right)+1\right\} / z\right] .\right]
\]