Lemma 4.25 . [0461]

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Lemma 4.25.

The explicit formula for $\beta_{p3}+\beta_{p4}$ is

\beta_{p3}+\beta_{p4}=\frac{-2\pi ie^{2\pi i\eta_{p}}}{1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}}+K_{p}(a)=\frac{-2\pi ie^{2\pi i\eta_{p}}}{f_{S}}+K_{p}(a),\quad p=1,2.

where the constant $K_{p}(a)$ is

\begin{split}K_{p}(a)=&\frac{\sqrt{-1}(a_{p\bar{2}}\text{Re}(a_{1\bar{2}})-a_{p\bar{1}}a_{2\bar{2}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{2\bar{2}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}}))\\ +&\frac{\sqrt{-1}(a_{p\bar{1}}\text{Re}(a_{1\bar{2}})-a_{p\bar{2}}a_{1\bar{1}})}{A}(\frac{\pi}{2}+\arctan(\frac{a_{1\bar{1}}+\text{Re}(a_{1\bar{2}})}{\sqrt{\mathbb{A}}})).\end{split}