In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show then that
$$\psi_n(h) \underset{n \rightarrow +\infty}{\longrightarrow} \ln\left(\mathrm{e}^{\beta} \operatorname{ch}(h) + \sqrt{\mathrm{e}^{2\beta} \operatorname{ch}^2(h) - 2\operatorname{sh}(2\beta)}\right).$$