Consider a 2-dimensional surface integral over the boundary, $\partial V$ of some region $V$ in 3-dimensional Euclidean space,
\begin{align} \mathcal{I}_{\partial V}(\vec{x})\equiv \int_{\partial V} d\vec{\Sigma}_2(\vec{y}) \cdot\vec{f}(\vec{y},\vec{x}), \end{align} where $d\vec{\Sigma}_2(\vec{y})$ is meant to denote the surface element on $\partial V$.
On the surface, suppose $\vec{f}$ is analytic in $\vec{y}$, $\vec{x}$ and $\vec{y}-\vec{x}$.
Question: Under what conditions will the surface integral $\mathcal{I}_{\partial V}(\vec{x})$ be either analytic in $\vec{x}$ or asymptotic to a power series in $\vec{x}$? By power series in $\vec{x}$, I mean \begin{align} \mathcal{I}_{\partial V}(\vec{x})=\sum_{\alpha\in\mathbb{N}_0^3}\frac{c_\alpha}{\alpha!}x^\alpha, \end{align} where $\alpha$ is a multi-index so $\alpha!\equiv\prod_{i=1}^3 \alpha_i$.