Proposition 3.3 . [0428]

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Proposition 3.3.

(Leading order asymptote) The formulae for $\bar{\alpha}_{i}$ are explicitly given as

(3.6)

\begin{cases}\bar{\alpha}_{1}=&\frac{1}{2\sqrt{a_{22}}}\{\log 2-\gamma_{E}-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}-\frac{\sqrt{a_{22}}}{\sqrt{A}}(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1})\right)\}\\ \bar{\alpha}_{2}=&\frac{1}{2\sqrt{a_{11}}}\{\log 2-\gamma_{E}-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}-\frac{\sqrt{a_{11}}}{\sqrt{A}}(\mu_{1}+\frac{a_{12}}{a_{11}}\mu_{2})\right)\}\\ \bar{\alpha}_{3}=&\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}}\{\log 2-\gamma_{E}\\ &-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}+\frac{a_{11}\mu_{1}+a_{12}\mu_{2}+a_{21}\mu_{1}+a_{22}\mu_{2}}{\sqrt{A}\sqrt{a_{11}+2a_{12}+a_{22}}}\right)\}\end{cases}

where $\gamma_{E}=\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k}-\log n$ is the Euler constant.