We focus on the neighbourhood of $\mathfrak{D}_{1}$. Modulo smooth terms

$$\alpha_{1}\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}},\quad\frac{\partial\alpha_{1}}{\partial\eta}\sim-\frac{a_{22}\bar{\eta}}{4({\mu_{1}^{2}+a_{22}|\eta|^{2}})^{3/2}},$$

so by the integral definitions, along $\mathfrak{D}_{1}$ the function $\beta_{2}$ is non-singular, and

$$\beta_{1}\sim\frac{-1}{2\eta}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}-1),\quad\beta_{0}\sim\frac{1}{\eta}-\beta_{1}=\frac{1}{2\eta}(\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+1).$$

Now

$$\begin{split}&d\log|z_{1}|=(a_{11}+v^{11})d\mu_{1}+(a_{12}+v^{12})d\mu_{2}+\text{Re}(\beta_{1}d\eta)\\ &\sim\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}d\mu_{1}+\frac{1}{2}(1-\frac{\mu_{1}}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}})d\log|\eta|+a_{11}d\mu_{1}+a_{12}d\mu_{2}\\ &=\frac{1}{2}d\log|\eta|+\frac{1}{2}d\sinh^{-1}(\frac{\mu_{1}}{\sqrt{a_{22}}|\eta|})+d(a_{11}\mu_{1}+a_{12}\mu_{2}),\end{split}$$

hence up to multiplying by a smooth function

$$|z_{1}|\sim\text{const}\cdot(\frac{\mu_{1}}{\sqrt{a_{22}}}+\sqrt{\frac{\mu_{1}^{2}+a_{22}|\eta|^{2}}{{a_{22}}}})^{1/2}e^{a_{11}\mu_{1}+a_{12}\mu_{2}}\to 0$$

as the point moves to $\mathfrak{D}_{1}$. Similarly

$$|z_{0}|\sim\text{const}\cdot(-\frac{\mu_{1}}{\sqrt{a_{22}}}+\sqrt{\frac{\mu_{1}^{2}+a_{22}|\eta|^{2}}{{a_{22}}}})^{1/2}e^{-a_{11}\mu_{1}-a_{12}\mu_{2}-a_{21}\mu_{1}-a_{22}\mu_{2}}\to 0.$$

The function $\log z_{2}$ encounters no singularity along $\mathfrak{D}_{1}$. These calculations guarantee the continuous extension of the holomorphic functions $z_{0},z_{1},z_{2}$ over $\mathfrak{D}_{1}$. Since the complex structure is compatible with the smooth topology by Section 2.3, these holomorphic functions in fact extend smoothly along $\mathfrak{D}_{1}$. We remark that what happens in these calculations is essentially identical to the Taub-NUT metric near the origin. ∎