This is problem on the integrability of a 2-form in the nhbd of its singularity.
I was looking at the general Cauchy formula (general because it works for $\mathcal C^1$ function, and makes the case of holomorphic ones descend from it) pasted below.
My problem is in the middle of the proof, when the book says that $$ d\omega=\frac{\partial_{\bar{\zeta}}g(\zeta)}{\zeta-z}d\zeta\wedge d\bar{\zeta} $$ is integrable at its singular point $z$. Why?
I can deduce what I'm asking for, from $(1.2.3)$ simply taking the first integral of RHS to LHS and passing to the limit LHS (which works because of $\frac1{2\pi i}\int_{\partial\Delta^{\varepsilon}}\omega\stackrel{\varepsilon\to0}{\to} g(z)$ as pointed at the end of the proof): in this way I could deduce that $d\omega$ is integrable at its singularity.
We notice that the book says this in a way which makes think that the integrability must descend from some very properties of $d\omega$ ("since $d\omega$ is integrable in its singularity $z$, then ..."), not in the way I did.
My question is: how? Maybe exploiting some properties of $d\omega$?
Can someone give me an hint? Thank you all.
