Lemma 4.11 . [0453]

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Lemma 4.11.

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

(4.17)

\begin{cases}\bar{\bar{\gamma}}_{1}=&-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{2\bar{2}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{2}-\frac{\text{Re}(a_{1\bar{2}})}{a_{2\bar{2}}}y_{1})\\ \bar{\bar{\gamma}}_{2}=&-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log(\frac{\mathbb{A}^{1/2}}{\sqrt{Aa_{1\bar{1}}}}|(y_{1},y_{2},\mu)|_{a}^{\prime}-y_{1}-\frac{\text{Re}(a_{1\bar{2}})}{a_{1\bar{1}}}y_{2})\\ \bar{\bar{\gamma}}_{3}=&-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\\ &\log(\frac{\mathbb{A}^{1/2}}{\sqrt{A(a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}})}}|(y_{1},y_{2},\mu)|_{a}^{\prime}+\frac{a_{1\bar{1}}y_{1}+\text{Re}(a_{1\bar{2}})y_{2}+\text{Re}(a_{2\bar{1}})y_{1}+a_{2\bar{2}}y_{2}}{a_{1\bar{1}}+2\text{Re}(a_{1\bar{2}})+a_{2\bar{2}}}).\end{cases}