Every holomorphic function on a homologically simply connected domain has a primitive

Given: A homologically simply connected complex domain $\mathrm{\Omega }$ and a holomorphic $f:\mathrm{\Omega }\to \mathbb{C}$.

If $\mathrm{\Gamma }$ is a cycle with trace in an open $\mathrm{\Omega }$, null-homologous in $\mathrm{\Omega }$, and $f$ is holomorphic on $\mathrm{\Omega }$, then ${\int }_{\mathrm{\Gamma }}fdz=0$ (Cauchy's theorem for a null-homologous cycle).

A complex domain is homologically simply connected when every cycle with trace in it is null-homologous in it (Homologically simply connected complex domains, Null-homologous cycles and homologous cycles in an open set).

A list of closed complex contours is a cycle; in particular a single closed contour, taken as the list of length $1$ with coefficient $1$, is a cycle, and its trace is the trace of that contour (Complex chains, their traces, and cycles).

For a chain consisting of the single closed contour $\gamma$ with coefficient $1$, ${\int }_{\mathrm{\Gamma }}fdz={\int }_{\gamma }fdz$ (Integration over a complex chain and the index of a chain).

For a complex domain $U$ and a continuous $f:U\to \mathbb{C}$, the following are equivalent: $f$ has a primitive on $U$; the integral of $f$ along rectifiable contours in $U$ depends only on the endpoints; the integral of $f$ around every closed rectifiable contour in $U$ is $0$ (For a continuous function on a complex domain, endpoint independence, zero closed-contour integrals, and existence of a primitive are equivalent).

A complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

A complex domain is a nonempty, connected, open subset of $\mathbb{C}$ (A complex domain is a nonempty connected open subset of $\mathbb{C}$).