Corollary 4.9 . [0450]

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Corollary 4.9.

(Green’s representation formula for $\bar{\gamma}_{i}$ )

\begin{cases}\bar{\gamma}_{1}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})-\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{2}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}<\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{1}^{\prime}\wedge(d\bar{\eta}_{1}^{\prime}-d\bar{\eta}_{2}^{\prime})-\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \bar{\gamma}_{3}(y_{1},y_{2},\mu)=&\text{Re}\lim_{\Lambda\to\infty}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S\cap\{y_{1}^{\prime}>-\Lambda\}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}-\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}\\ \bar{\gamma}_{4}(y_{1},y_{2},\mu)=&\text{Im}\{\frac{A^{1/2}}{4\sqrt{\mathbb{A}}}\int_{S}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\}.\end{cases}