Cauchy's theorem for a null-homologous cycle

Given: An open $\mathrm{\Omega }$, a holomorphic $f:\mathrm{\Omega }\to \mathbb{C}$, and a cycle $\mathrm{\Gamma }$ with ${\mathrm{\Gamma }}^{\ast }\subseteq \mathrm{\Omega }$ which is null-homologous in $\mathrm{\Omega }$; the plane is read as ${\mathbb{R}}^2$ through $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves.

Under the hypotheses above, $n(\mathrm{\Gamma },z)f(z)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}f(\zeta )(\zeta -z{)}^{-1}d\zeta$ for every $z\in \mathrm{\Omega }\setminus {\mathrm{\Gamma }}^{\ast }$ (Cauchy's integral formula for a null-homologous cycle).

Products of functions complex differentiable at a point are complex differentiable there, and constants have derivative $0$ (Linearity, product, reciprocal, and quotient rules for complex derivatives); a complex differentiable function is continuous (Complex differentiability at a point implies continuity there).

${\int }_{\mathrm{\Gamma }}fdz={\sum }_{k<r,m_k\mathrm{\ne }0}{m}_k{\int }_{{\gamma }_k}fdz$, and $n(\mathrm{\Gamma },z)=(2\pi i{)}^{-1}{\int }_{\mathrm{\Gamma }}d\zeta \mathrm{/}(\zeta -z)$ for $z\mathrm{\notin }{\mathrm{\Gamma }}^{\ast }$ (Integration over a complex chain and the index of a chain); a chain is a finite list of integer-weighted complex contours and its trace is the union of the ${\gamma }_k^{\ast }$ with $m_k\mathrm{\ne }0$ (Complex chains, their traces, and cycles).

A cycle $\mathrm{\Gamma }$ with trace in $\mathrm{\Omega }$ is null-homologous in $\mathrm{\Omega }$ when $n(\mathrm{\Gamma },p)=0$ for every $p\in \mathbb{C}\setminus \mathrm{\Omega }$ (Null-homologous cycles and homologous cycles in an open set).

Complex line integrals are linear in the integrand (Complex line integrals are linear in the integrand).

A compact subset is closed and bounded (A compact subset of a metric space is closed and bounded); a finite union of compact subsets is compact (A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact); a continuous image of a compact subset is compact (The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset); a closed bounded interval is compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

A topological space is connected exactly when its only clopen subsets are the empty set and the whole space (For a topological space the following agree: no separation exists, the only clopen subsets are $\mathrm{\varnothing }$ and $X$, and every continuous map to the two-point discrete space is constant).

For $n\ge 1$, ${\mathbb{R}}^n$ is polygonally connected and connected (${\mathbb{R}}^n$ is polygonally connected, connected, locally path-connected and locally connected).

A set is closed exactly when its complement is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement), and a subset is bounded when it is empty or lies inside a ball (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space).