Let $D$ be a bounded open subset of $\mathbb{C}$.
Let $u$ be a smooth function on $\mathbb{C}$ supported on a compact subset of $\Delta ( \xi_0, \delta) \equiv \{ |\xi-\xi_0| < \delta\}$ such that $u=1$ in a neighborhood of the closure of $\Delta(\xi_0, \delta/2)$.
Set $f = 0$ off $D$, and let $G$ be the solution of the $\bar{\partial}$-equation $\bar{\partial} G = f \bar{\partial} u$ which vanishes at $\infty$. The function $G$ is given by explicitly by $$ G(\xi) = f(\xi) u(\xi) + \frac{1}{\pi} \iint f(\lambda) \frac{\partial u}{\partial \bar{\lambda}} \frac{1}{\lambda-\xi}\, d\zeta d\eta, $$ where $\lambda = \zeta + i \eta$.
I have no idea how this $G$ is obtained. If there is a book which contains this type of thing, can anyone recommend such book?
Any help will be appreciated!