I'm trying to understand whether a certain integral representation is holomorphic. Specifically, let's assume that $f$ belongs to the Schwartz space $\mathcal S(\mathbb R)$, and consider a function $g(z)$ represented in the following way:
\begin{align*} g(z) = \int_{\mathbb R} \frac{f(y)}{z- y} dy \end{align*}
It is my understanding that this function should be holomorphic when $\textrm{Im z} > 0$ and $\textrm{Im z} < 0$. This is because the integrand $\frac{f(y)}{z- y}$ is holomorphic in those regions. This can be shown as follows:
\begin{align*} \partial_{\bar z} \int_{\mathbb R} \frac{f(y)}{z- y} dy = \int_{\mathbb R} \partial_{\bar z} \frac{f(y)}{z- y} dy = 0 \end{align*}
So it seems that the function is indeed holomorphic. However, I'm not entirely sure of this reasoning. Is there anything wrong with it, or is there something else I should consider?
Any help and explanations would be greatly appreciated. Thanks in advance.