Holomorphic primitive of holomorphic function in several complex variables

Question

Given a holomorphic function $f:\Omega\subset\mathbb{C}^n\rightarrow\mathbb{C}$,with $\Omega$ open and connected, when is it possible to find a holomorphic $m-$primitive, i.e. a holomorphic function (in every variable) $g:\Omega\rightarrow\mathbb{C}$ such that \begin{equation} f=\dfrac{\partial g}{\partial z_m}? \end{equation} Maybe, it is simply a consequence of the 1-variable version of the problem of finding a primitive of a holomorphic function, where it is enough to assume $\Omega$ simply connected, so in the several variables version, if one define \begin{equation} \Omega_m:=\{w\in\mathbb{C}\mid (z_1,...,z_{m-1},w,z_{m+1},...,z_n)\in\Omega\} \end{equation} then if $\Omega_m$ is simply connected, such an $m-$primitive $g$ exists? Can someone give me a rather formal proof? Thank you in advance.

Answer 1

My attempt.
Define $g:\Omega\rightarrow\mathbb{C}$ by \begin{equation} g(z_1,...,z_n):=\int_{z^0_m}^{z_m}f(z_1,...,z_{m-1},w,z_{m+1},...,z_n)dw, \end{equation} where $z^0=(z^0_1,...,z^0_n)$ is any point in $\Omega$.
Note that $g$ is well defined since it does not depend on the choice of the path between $z^0_m$ and $z_m$. Indeed, let $\gamma_1$ and $\gamma_2$ any two paths between $z^0_m$ and $z_m$, then $\gamma_1\cup -\gamma_2$ is closed and since $\Omega_m$ is simply connected, Cauchy's Theorem applied to the holomorphic function $f$ guarantees that $$\int_{\gamma_1}f-\int_{\gamma_2}f=0.$$ Now, by the fundamental theorem of calculus it holds $\dfrac{\partial g}{\partial x_m}=f$, moreover $g$ is holomorphic, since $$\dfrac{\partial g}{\partial{\bar{z}_j}}=\int_{z^0_m}^{z_m}\dfrac{\partial f}{\partial{\bar{z}_j}}dw=0.$$