Let $\mu_{1},\mu_{2}$ be fixed. The essential task is to understand the asymptotic behaviour of $\beta_{i}(\mu_{1},\mu_{2},\eta+n)$ as $|n|$ becomes large. We focus on $\beta_{1}$.

Using the homogeneity property of $\alpha_{1},\alpha_{2},\alpha_{3}$ in the $\mu_{1},\mu_{2}$ and $\eta$ variables, it is easy to see from the integral definition of $\beta_{1}$ that

$$\beta_{1}(0,0,\eta)=\frac{1}{\eta}\beta_{1}(0,0,1),\quad|\beta_{1}(0,0,1)|\leq C.$$

By elementary properties of arctan

$$\alpha_{1}(\mu_{1},\mu_{2},\eta)=\frac{1}{4\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+O(\frac{|\mu_{1}|+|\mu_{2}|}{\mu_{1}^{2}+a_{22}|\eta|^{2}}),$$

$$\frac{\partial\alpha_{1}}{\partial\eta}=\frac{-a_{22}\bar{\eta}}{8(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{1/2}})\right),$$

and similarly

$$\frac{\partial\alpha_{3}}{\partial\eta}=\frac{-(a_{11}+2a_{22}+a_{22})\bar{\eta}}{8((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{22}+a_{22})|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{((\mu_{1}-\mu_{2})^{2}+A^{1/2}|\eta|^{2})^{1/2}})\right).$$

After integration

$$\begin{split}|\beta_{1}(\mu_{1},\mu_{2},\eta)-\beta_{1}(0,0,\eta)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{|\eta|}(\frac{1}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{1}{\sqrt{(\mu_{1}-\mu_{2})^{2}+a_{22}|\eta|^{2}}}).\end{split}$$

This shows the series

$$\sum_{n\in\mathbb{Z}}\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)$$

is absolutely convergent if $\eta\notin\mathbb{Z}$, and if morever $|x|\leq\frac{1}{2}$ then we have the bound

$$\sum_{n\in\mathbb{Z}\setminus\{0\}}|\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{A^{1/4}}.$$

Thus the convergence of the series $\tilde{\beta}_{1}$ is equivalent to the convergence of

$$\lim_{N\to\infty}\sum_{n=-N}^{N}\beta_{1}(0,0,\eta+n)=\beta_{1}(0,0,1)\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{\eta+n}=\beta_{1}(0,0,1)\pi\cot(\pi\eta),$$

and similarly for $\tilde{\beta}_{2}$ and $\tilde{\beta}_{0}$. The periodicity claim follows from standard rearranging theorems for series. The estimate on $\tilde{\beta}_{i}-\beta_{i}$ follows by combining the above discussions. ∎