Lemma 4.24 . [045Z]

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Lemma 4.24.

(Asymptotes as $\mu\to\pm\infty$ ) For any fixed $\eta_{1},\eta_{2}$ ,

\begin{cases}\lim_{\mu\to+\infty}\beta_{p3}(\eta_{1},\eta_{2},\mu)=0,\\ \lim_{\mu\to-\infty}\beta_{p4}(\eta_{1},\eta_{2},\mu)=0.\end{cases}

Morever

\begin{cases}\lim_{\mu\to+\infty}\frac{\partial\beta_{p3}}{\partial\bar{\eta}_{p}}(\eta_{1},\eta_{2},\mu)=0,\\ \lim_{\mu\to-\infty}\frac{\partial\beta_{p4}}{\partial\bar{\eta}_{p}}(\eta_{1},\eta_{2},\mu)=0.\end{cases}