Let $U \subset \subset \mathbb{C}$ (that is, $\bar U \subset \mathbb{C}$ is compact) and let $f$ be a complex-valued function, which is continuously differentiable in a neighbourhood of $\bar U$.
Consider the integral \begin{align*} u(z) := \int\limits_U \frac{f(\xi)}{\xi - z} d\xi \wedge d\bar \xi, ~~~~~ \text{where} ~ z \in U. \end{align*}
My question is: Why does this integral always exist? -- as the integrand is not bounded.
To give some context: $u$ (divided by $2\pi i$) is the solution of the inhomogeneous Cauchy-Riemann equation $\frac{\partial u}{\partial \bar z} = f$.
Many thanks in advance.