\right\vert }\right) ^{\frac{\varepsilon }{1-\varepsilon }}.  \label{eq6iv.3}\end{aligned}$$

Splitting the argument of the exponential in same way as and using , we have $$\begin{aligned}
\left\vert {\operatorname{Im}}\lambda \right\vert \cdot x_{n}^{\varepsilon }-\frac{\widetilde{c}}{r^{\alpha }}x_{n}^{\alpha }& \leq \left\vert {\operatorname{Im}}\lambda
\right\vert \cdot x_{n}^{\varepsilon }\left( \frac{\widetilde{c}r^{-\varepsilon -1}}{\left\vert {\operatorname{Im}}\lambda \right\vert }-\frac{r^{-\varepsilon }\delta \widetilde{c}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\right) -(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }} \\
& =-x_{n}^{\varepsilon }\widetilde{c}r^{-\varepsilon -1}(r\delta
-1)-(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }} \\
& \leq -\left( \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\right) ^{\frac{\varepsilon }{1+\varepsilon }}\widetilde{c}r^{-\varepsilon -1}(r\delta -1)-(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }} \\
& -\frac{\widetilde{c}^{\frac{1}{1-\varepsilon }}}{r^{\frac{2\varepsilon +1}{1-\varepsilon }}}\frac{r\delta -1}{\left\vert {\operatorname{Im}}\lambda \right\vert ^{\frac{\varepsilon }{1-\varepsilon }}}-(1-\delta )\frac{\widetilde{c}x_{n}^{\varepsilon }}{r^{\varepsilon }}.\end{aligned}$$

Inserting this into we obtain $$\begin{aligned}
\left\vert \partial _{\overline{\lambda }}\widetilde{G}\right\vert &
\lesssim _{g}\left( \sum_{n\geq 0}x_{n}^{1-\varepsilon }\exp \{-\limfunc{const}x_{n}^{\varepsilon }\}\right) \cdot \exp \left\{ -\widetilde{Q}\left\vert {\operatorname{Im}}\lambda \right\vert ^{-\frac{\varepsilon }{1-\varepsilon }}\right\}  \label{eq6v.0} \\
\widetilde{Q}& \equiv(r\delta -1)\frac{\widetilde{c}^{\frac{1}{1-\varepsilon 
}}}{r^{\frac{2\varepsilon +1}{1-\varepsilon }}}<(r\delta -1)\frac{c^{\frac{1}{1-\varepsilon }}}{r^{\frac{2\varepsilon +1}{1-\varepsilon }}}.
\label{eq6v.1}\end{aligned}$$The series in is convergent and $\widetilde{G}\left( \lambda
\right) $ is an pseudo analytic extension of $\widehat{g}\left( \lambda
\right) $ from the real line to the upper half plane. Due to we have found a pseudo analytic extension of $G$ subject to with $Q=2\widetilde{Q}$. This completes our proof.

The representation is not unique. It depends on the reference point in the splitting of . This flexibility will be used 